In mathematics, many logarithmic identities exist. The following is a compilation of the notable of these, many of which are used for computational purposes.
Trivial identities
Trivial mathematical identities are relatively simple (for an experienced mathematician), though not necessarily unimportant. Trivial logarithmic identities are:
![{\displaystyle \log _{b}(1)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/901f6efd3f7b26aa95b855e884a8c2c620ef1fe0) | because | |
![{\displaystyle \log _{b}(b)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3a58a8d06818394825efc588fa84970424b75f8) | because | |
Explanations
By definition, we know that:
,
where
and
.
Setting
, we can see that:
. So, substituting these values into the formula, we see that:
, which gets us the first property.
Setting
, we can see that:
. So, substituting these values into the formula, we see that:
, which gets us the second property.
Cancelling exponentials
Logarithms and exponentials with the same base cancel each other. This is true because logarithms and exponentials are inverse operations—much like the same way multiplication and division are inverse operations, and addition and subtraction are inverse operations.
![{\displaystyle b^{\log _{b}(x)}=x{\text{ because }}{\mbox{antilog}}_{b}(\log _{b}(x))=x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1cfe924e512bd3ca53cf347d63f7d3f7272fb41)
[1]
Both of the above are derived from the following two equations that define a logarithm: (note that in this explanation, the variables of
and
may not be referring to the same number)
![{\displaystyle \color {black}\log \color {blue}_{b}\color {black}(\color {green}y\color {black})=\color {red}x\color {black}\iff \color {blue}b\color {black}\color {red}^{x}\color {black}=\color {green}y\color {black}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf527d15865729252dc514eb98a807fadf4887e)
Looking at the equation
, and substituting the value for
of
, we get the following equation:
, which gets us the first equation. Another more rough way to think about it is that
, and that that "
" is
.
Looking at the equation
, and substituting the value for
of
, we get the following equation:
, which gets us the second equation. Another more rough way to think about it is that
, and that that something "
" is
.
Using simpler operations
Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below.[2] The first three operations below assume that x = bc and/or y = bd, so that logb(x) = c and logb(y) = d. Derivations also use the log definitions x = blogb(x) and x = logb(bx).
![{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a72b4b7ba4c487ba5c15587d2eff610355605901) | because | |
![{\displaystyle \log _{b}({\tfrac {x}{y}})=\log _{b}(x)-\log _{b}(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3376dd2b0c1a700e1e60f7897b953ba52c696fb) | because | |
![{\displaystyle \log _{b}(x^{d})=d\log _{b}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d23edf2cabd7544f17387e50fbad8ce772cdedad) | because | |
![{\displaystyle \log _{b}\left({\sqrt[{y}]{x}}\right)={\frac {\log _{b}(x)}{y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de242874c347b5ca76ff4594f7595f5c94ff935e) | because | |
![{\displaystyle x^{\log _{b}(y)}=y^{\log _{b}(x)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f223e2054ba145e70ed80ffbc4ccc7ff59bc7479) | because | |
![{\displaystyle c\log _{b}(x)+d\log _{b}(y)=\log _{b}(x^{c}y^{d})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a65199c28cefee092329b6d7617a6e3c1531ac) | because | |
Where
,
, and
are positive real numbers and
, and
and
are real numbers.
The laws result from canceling exponentials and the appropriate law of indices. Starting with the first law:
![{\displaystyle xy=b^{\log _{b}(x)}b^{\log _{b}(y)}=b^{\log _{b}(x)+\log _{b}(y)}\Rightarrow \log _{b}(xy)=\log _{b}(b^{\log _{b}(x)+\log _{b}(y)})=\log _{b}(x)+\log _{b}(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd9191539d435c6eb56105337bcbef3ae0edc6a)
The law for powers exploits another of the laws of indices:
![{\displaystyle x^{y}=(b^{\log _{b}(x)})^{y}=b^{y\log _{b}(x)}\Rightarrow \log _{b}(x^{y})=y\log _{b}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9071d5658996d138963a3462d5c536b53cf347f)
The law relating to quotients then follows:
![{\displaystyle \log _{b}{\bigg (}{\frac {x}{y}}{\bigg )}=\log _{b}(xy^{-1})=\log _{b}(x)+\log _{b}(y^{-1})=\log _{b}(x)-\log _{b}(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b66def1e61152706cd9441e03c42aec2dc5a386c)
![{\displaystyle \log _{b}{\bigg (}{\frac {1}{y}}{\bigg )}=\log _{b}(y^{-1})=-\log _{b}(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b4002f27fcdb9c71e363e41deffb9cbba7fb828)
Similarly, the root law is derived by rewriting the root as a reciprocal power:
![{\displaystyle \log _{b}({\sqrt[{y}]{x}})=\log _{b}(x^{\frac {1}{y}})={\frac {1}{y}}\log _{b}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d3eba91cc156e4939f7136dc81d43c2f391e449)
Derivations of product, quotient, and power rules
These are the three main logarithm laws/rules/principles,[3] from which the other properties listed above can be proven. Each of these logarithm properties correspond to their respective exponent law, and their derivations/proofs will hinge on those facts. There are multiple ways to derive/prove each logarithm law – this is just one possible method.
Logarithm of a product
To state the logarithm of a product law formally:
![{\displaystyle \forall b\in \mathbb {R} _{+},b\neq 1,\forall x,y,\in \mathbb {R} _{+},\log _{b}(xy)=\log _{b}(x)+\log _{b}(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1819bbd4c12965c0f2be47c0deda25373a81ca2)
Derivation:
Let
, where
, and let
. We want to relate the expressions
and
. This can be done more easily by rewriting in terms of exponentials, whose properties we already know. Additionally, since we are going to refer to
and
quite often, we will give them some variable names to make working with them easier: Let
, and let
.
Rewriting these as exponentials, we see that
![{\displaystyle {\begin{aligned}m&=\log _{b}(x)\iff b^{m}=x,\\n&=\log _{b}(y)\iff b^{n}=y.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5544c72a450e8c629e0755d33c6b9f404c6aa37)
From here, we can relate
(i.e.
) and
(i.e.
) using exponent laws as
![{\displaystyle xy=(b^{m})(b^{n})=b^{m}\cdot b^{n}=b^{m+n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db0436e02a3035580e2242b7d57a4843797230c1)
To recover the logarithms, we apply
to both sides of the equality.
![{\displaystyle \log _{b}(xy)=\log _{b}(b^{m+n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fed206e900c4a6dc9c10b0078345ba6e5c34ccc)
The right side may be simplified using one of the logarithm properties from before: we know that
, giving
![{\displaystyle \log _{b}(xy)=m+n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80a8d1e3c2fbcfbb10f5f37c98a546d8dd17cc52)
We now resubstitute the values for
and
into our equation, so our final expression is only in terms of
,
, and
.
![{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a72b4b7ba4c487ba5c15587d2eff610355605901)
This completes the derivation.
Logarithm of a quotient
To state the logarithm of a quotient law formally:
![{\displaystyle \forall b\in \mathbb {R} _{+},b\neq 1,\forall x,y,\in \mathbb {R} _{+},\log _{b}\left({\frac {x}{y}}\right)=\log _{b}(x)-\log _{b}(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2005fd0201bf9f8f2abe66f46edf601624184597)
Derivation:
Let
, where
, and let
.
We want to relate the expressions
and
. This can be done more easily by rewriting in terms of exponentials, whose properties we already know. Additionally, since we are going to refer to
and
quite often, we will give them some variable names to make working with them easier: Let
, and let
.
Rewriting these as exponentials, we see that:
![{\displaystyle {\begin{aligned}m&=\log _{b}(x)\iff b^{m}=x,\\n&=\log _{b}(y)\iff b^{n}=y.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5544c72a450e8c629e0755d33c6b9f404c6aa37)
From here, we can relate
(i.e.
) and
(i.e.
) using exponent laws as
![{\displaystyle {\frac {x}{y}}={\frac {(b^{m})}{(b^{n})}}={\frac {b^{m}}{b^{n}}}=b^{m-n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/150388d9b5e2205bc3939075465bdcff9b9a74f0)
To recover the logarithms, we apply
to both sides of the equality.
![{\displaystyle \log _{b}\left({\frac {x}{y}}\right)=\log _{b}\left(b^{m-n}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8766a4e082871462a6f75ff809b01831479c0c9)
The right side may be simplified using one of the logarithm properties from before: we know that
, giving
![{\displaystyle \log _{b}\left({\frac {x}{y}}\right)=m-n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1beb8e712b8d231f95e1dd95d363f167100d835f)
We now resubstitute the values for
and
into our equation, so our final expression is only in terms of
,
, and
.
![{\displaystyle \log _{b}\left({\frac {x}{y}}\right)=\log _{b}(x)-\log _{b}(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80a1fab73bd105f35aa61fc3956ea758d9b7df47)
This completes the derivation.
Logarithm of a power
To state the logarithm of a power law formally:
![{\displaystyle \forall b\in \mathbb {R} _{+},b\neq 1,\forall x\in \mathbb {R} _{+},\forall r\in \mathbb {R} ,\log _{b}(x^{r})=r\log _{b}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b2653f1445d64d6c9494e386fee7a916988ce2)
Derivation:
Let
, where
, let
, and let
. For this derivation, we want to simplify the expression
. To do this, we begin with the simpler expression
. Since we will be using
often, we will define it as a new variable: Let
.
To more easily manipulate the expression, we rewrite it as an exponential. By definition,
, so we have
![{\displaystyle b^{m}=x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d0cdf146015ff5de88f27856c0714898d03a82)
Similar to the derivations above, we take advantage of another exponent law. In order to have
in our final expression, we raise both sides of the equality to the power of
:
![{\displaystyle {\begin{aligned}(b^{m})^{r}&=(x)^{r}\\b^{mr}&=x^{r}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d786d3558eeab7c89fabf566216dc2070464e541)
where we used the exponent law
.
To recover the logarithms, we apply
to both sides of the equality.
![{\displaystyle \log _{b}(b^{mr})=\log _{b}(x^{r})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32d53bc8488c2b72d78492cfcc2b5e7f0e001772)
The left side of the equality can be simplified using a logarithm law, which states that
.
![{\displaystyle mr=\log _{b}(x^{r})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/534ff0df9a6bef3663e19ffc3b8a6df84ef7b499)
Substituting in the original value for
, rearranging, and simplifying gives
![{\displaystyle {\begin{aligned}\left(\log _{b}(x)\right)r&=\log _{b}(x^{r})\\r\log _{b}(x)&=\log _{b}(x^{r})\\\log _{b}(x^{r})&=r\log _{b}(x)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f7e538ecbb3d4c01cb2af97ee1d393a48233af0)
This completes the derivation.
Changing the base
To state the change of base logarithm formula formally:
This identity is useful to evaluate logarithms on calculators. For instance, most calculators have buttons for ln and for log10, but not all calculators have buttons for the logarithm of an arbitrary base.
Proof/derivation
Let
, where
Let
. Here,
and
are the two bases we will be using for the logarithms. They cannot be 1, because the logarithm function is not well defined for the base of 1.[citation needed] The number
will be what the logarithm is evaluating, so it must be a positive number. Since we will be dealing with the term
quite frequently, we define it as a new variable: Let
.
To more easily manipulate the expression, it can be rewritten as an exponential.
Applying
to both sides of the equality,
Now, using the logarithm of a power property, which states that
,
Isolating
, we get the following:
Resubstituting
back into the equation,
This completes the proof that
.
This formula has several consequences:
where
is any permutation of the subscripts 1, ..., n. For example
Summation/subtraction
The following summation/subtraction rule is especially useful in probability theory when one is dealing with a sum of log-probabilities:
| because | |
| because | |
Note that the subtraction identity is not defined if
, since the logarithm of zero is not defined. Also note that, when programming,
and
may have to be switched on the right hand side of the equations if
to avoid losing the "1 +" due to rounding errors. Many programming languages have a specific log1p(x)
function that calculates
without underflow (when
is small).
More generally:
Exponents
A useful identity involving exponents:
or more universally:
Other/resulting identities
Inequalities
Based on,[4][5] and [6]
![{\displaystyle {\frac {x}{1+x}}\leq \ln(1+x)\leq {\frac {x(6+x)}{6+4x}}\leq x{\mbox{ for all }}{-1}<x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cac772d43c9e3b130cf2cfdfc0b792fe4275fef8)
![{\displaystyle {\begin{aligned}{\frac {2x}{2+x}}&\leq 3-{\sqrt {\frac {27}{3+2x}}}\leq {\frac {x}{\sqrt {1+x+x^{2}/12}}}\\[4pt]&\leq \ln(1+x)\leq {\frac {x}{\sqrt {1+x}}}\leq {\frac {x}{2}}{\frac {2+x}{1+x}}\\[4pt]&{\text{ for }}0\leq x{\text{, reverse for }}{-1}<x\leq 0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7129f5e2c3c1cd0dd7b0e5c1d7bf181d569539c5)
All are accurate around
, but not for large numbers.
Calculus identities
![{\displaystyle \lim _{x\to 0^{+}}\log _{a}(x)=-\infty \quad {\mbox{if }}a>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/050eb88efd0aa05ed0940a1ea2173b61ae5d2bb0)
![{\displaystyle \lim _{x\to 0^{+}}\log _{a}(x)=\infty \quad {\mbox{if }}0<a<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e19146c97b6d735e6903161940937f831d8d9cfe)
![{\displaystyle \lim _{x\to \infty }\log _{a}(x)=\infty \quad {\mbox{if }}a>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c79ee13d90275ff2ae78c27806be74fe07d74367)
![{\displaystyle \lim _{x\to \infty }\log _{a}(x)=-\infty \quad {\mbox{if }}0<a<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2887cfd0e0a0d3b0eb4a11a1ddd5648d3e1024c1)
![{\displaystyle \lim _{x\to 0^{+}}x^{b}\log _{a}(x)=0\quad {\mbox{if }}b>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3160b82b1f938dd09d8ad3ad345e65259e6911cd)
![{\displaystyle \lim _{x\to \infty }{\frac {\log _{a}(x)}{x^{b}}}=0\quad {\mbox{if }}b>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43b8207752a5377c71c1f53b40e203c0dd658ad6)
The last limit is often summarized as "logarithms grow more slowly than any power or root of x".
Derivatives of logarithmic functions
![{\displaystyle {d \over dx}\ln x={1 \over x},x>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50360c92e6ea322d60a834025dca84f2a465029e)
![{\displaystyle {d \over dx}\ln |x|={1 \over x},x\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2148e76bad2dd6be9bc243af52a3386a26000c)
![{\displaystyle {d \over dx}\log _{a}x={1 \over x\ln a},x>0,a>0,{\text{ and }}a\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50023becf3cda12c7672df825c1a168d1fbfa219)
Integral definition
![{\displaystyle \ln x=\int _{1}^{x}{\frac {1}{t}}\ dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0333112e8e5debcf0e84265a90583cc0fac3c65f)
Riemann Sum
![{\displaystyle \ln(n+1)=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3104e661fe73ecd7c5c36e76ecf85c061e872f52)
![{\displaystyle \lim _{k\to \infty }\sum _{i=1}^{k}{\frac {1}{x_{i}}}\Delta x=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0581b2fbdb5291a17de060dda31aa172d6208c65)
![{\displaystyle \lim _{k\to \infty }\sum _{i=1}^{k}{\frac {1}{1+{\frac {i-1}{k}}n}}\cdot {\frac {n}{k}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efcec08e87389c708f27bd6700ae6440e6514222)
![{\displaystyle \lim _{k\to \infty }\sum _{x=1}^{k\cdot n}{\frac {1}{1+{\frac {x}{k}}}}\cdot {\frac {1}{k}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b041f18bcc080c1a7866daa2a512c86f93a1a4)
![{\displaystyle \lim _{k\to \infty }\sum _{x=1}^{k\cdot n}{\frac {1}{k+x}}=\lim _{k\to \infty }\sum _{x=k+1}^{k\cdot n+k}{\frac {1}{x}}=\lim _{k\to \infty }\sum _{x=k+1}^{k(n+1)}{\frac {1}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff71d733b6b21bf5f39559a7ebbc8e6de0b500a7)
for
and
is a sample point in each interval.
Series representation
The natural logarithm
has a well-known Taylor series[7] expansion that converges for
in the open-closed interval
:
Within this interval, for
, the series is conditionally convergent, and for all other values, it is absolutely convergent. For
or
, the series does not converge to
. In these cases, different representations or methods must be used to evaluate the logarithm.
Harmonic number difference
It is not uncommon in advanced mathematics, particularly in analytic number theory and asymptotic analysis, to encounter expressions involving differences or ratios of harmonic numbers at scaled indices.[8] The identity involving the limiting difference between harmonic numbers at scaled indices and its relationship to the logarithmic function provides an intriguing example of how discrete sequences can asymptotically relate to continuous functions. This identity is expressed as[9]
![{\displaystyle \lim _{k\to \infty }(H_{k(n+1)}-H_{k})=\ln(n+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb5ac4d569376b15d632cb2ca6829ecdb38be4a6)
which characterizes the behavior of harmonic numbers as they grow large. This approximation (which precisely equals
in the limit) reflects how summation over increasing segments of the harmonic series exhibits integral properties, giving insight into the interplay between discrete and continuous analysis. It also illustrates how understanding the behavior of sums and series at large scales can lead to insightful conclusions about their properties. Here
denotes the
-th harmonic number, defined as
![{\displaystyle H_{k}=\sum _{j=1}^{k}{\frac {1}{j}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab3bb9ae2a7c76dccdfd310b421f371dde2a03a0)
The harmonic numbers are a fundamental sequence in number theory and analysis, known for their logarithmic growth. This result leverages the fact that the sum of the inverses of integers (i.e., harmonic numbers) can be closely approximated by the natural logarithm function, plus a constant, especially when extended over large intervals.[10][8][11] As
tends towards infinity, the difference between the harmonic numbers
and
converges to a non-zero value. This persistent non-zero difference,
, precludes the possibility of the harmonic series approaching a finite limit, thus providing a clear mathematical articulation of its divergence.[12][13] The technique of approximating sums by integrals (specifically using the integral test or by direct integral approximation) is fundamental in deriving such results. This specific identity can be a consequence of these approximations, considering:
![{\displaystyle \sum _{j=k+1}^{k(n+1)}{\frac {1}{j}}\approx \int _{k}^{k(n+1)}{\frac {dx}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15370b4fdb442a025d131103eca43e4aec0dc942)
Harmonic limit derivation
The limit explores the growth of the harmonic numbers when indices are multiplied by a scaling factor and then differenced. It specifically captures the sum from
to
:
![{\displaystyle H_{k(n+1)}-H_{k}=\sum _{j=k+1}^{k(n+1)}{\frac {1}{j}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef23eee16dc30fce984c449309fc80e92e96ef0a)
This can be estimated using the integral test for convergence, or more directly by comparing it to the integral of
from
to
:
![{\displaystyle \lim _{k\to \infty }\sum _{j=k+1}^{k(n+1)}{\frac {1}{j}}=\int _{k}^{k(n+1)}{\frac {dx}{x}}=\ln(k(n+1))-\ln(k)=\ln \left({\frac {k(n+1)}{k}}\right)=\ln(n+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d157a5857ba414a72574f99df4badf592f981b0)
As the window's lower bound begins at
and the upper bound extends to
, both of which tend toward infinity as
, the summation window encompasses an increasingly vast portion of the smallest possible terms of the harmonic series (those with astronomically large denominators), creating a discrete sum that stretches towards infinity, which mirrors how continuous integrals accumulate value across an infinitesimally fine partitioning of the domain. In the limit, the interval is effectively from
to
where the onset
implies this minimally discrete region.
The harmonic number difference formula for
is an extension[9] of the classic, alternating identity of
:
![{\displaystyle \ln(2)=\lim _{k\to \infty }\sum _{n=1}^{k}\left({\frac {1}{2n-1}}-{\frac {1}{2n}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/556d678809453a85dcf212dc9753f963ea7700fd)
which can be generalized as the double series over the residues of
:
![{\displaystyle \ln(m)=\sum _{x\in \langle m\rangle \cap \mathbb {N} }\sum _{r\in \mathbb {Z} _{m}\cap \mathbb {N} }\left({\frac {1}{x-r}}-{\frac {1}{x}}\right)=\sum _{x\in \langle m\rangle \cap \mathbb {N} }\sum _{r\in \mathbb {Z} _{m}\cap \mathbb {N} }{\frac {r}{x(x-r)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/951c9d3b22f46163e691376c25cee84ff88226fa)
where
is the principle ideal generated by
. Subtracting
from each term
(i.e., balancing each term with the modulus) reduces the magnitude of each term's contribution, ensuring convergence by controlling the series' tendency toward divergence as
increases. For example:
![{\displaystyle \ln(4)=\lim _{k\to \infty }\sum _{n=1}^{k}\left({\frac {1}{4n-3}}-{\frac {1}{4n}}\right)+\left({\frac {1}{4n-2}}-{\frac {1}{4n}}\right)+\left({\frac {1}{4n-1}}-{\frac {1}{4n}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75943d2e1d4f1db70bed4c5e49f95eb9e403882f)
This method leverages the fine differences between closely related terms to stabilize the series. The sum over all residues
ensures that adjustments are uniformly applied across all possible offsets within each block of
terms. This uniform distribution of the "correction" across different intervals defined by
functions similarly to telescoping over a very large sequence. It helps to flatten out the discrepancies that might otherwise lead to divergent behavior in a straightforward harmonic series.
Deveci's Proof
A fundamental feature of the proof is the accumulation of the subtrahends
into a unit fraction, that is,
for
, thus
rather than
, where the extrema of
are
if
and
otherwise, with the minimum of
being implicit in the latter case due to the structural requirements of the proof. Since the cardinality of
depends on the selection of one of two possible minima, the integral
, as a set-theoretic procedure, is a function of the maximum
(which remains consistent across both interpretations) plus
, not the cardinality (which is ambiguous[14][15] due to varying definitions of the minimum). Whereas the harmonic number difference computes the integral in a global sliding window, the double series, in parallel, computes the sum in a local sliding window—a shifting
-tuple—over the harmonic series, advancing the window by
positions to select the next
-tuple, and offsetting each element of each tuple by
relative to the window's absolute position. The sum
corresponds to
which scales
without bound. The sum
corresponds to the prefix
trimmed from the series to establish the window's moving lower bound
, and
is the limit of the sliding window (the scaled, truncated[16] series):
![{\displaystyle \sum _{n=1}^{k}\sum _{r=1}^{\omega }\left({\frac {1}{mn-r}}-{\frac {1}{mn}}\right)=\sum _{n=1}^{k}\sum _{r=0}^{\omega }\left({\frac {1}{mn-r}}-{\frac {1}{mn}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d5f65e60cb9293863cb922eb74889d9b692b5d3)
![{\displaystyle =\sum _{n=1}^{k}\left(-{\frac {1}{n}}+\sum _{r=0}^{\omega }{\frac {1}{mn-r}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aacefdd0081b9acd2ced7447e14dea646691eabb)
![{\displaystyle =-H_{k}+\sum _{n=1}^{k}\sum _{r=0}^{\omega }{\frac {1}{mn-r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c874b515ffedde162d985c08f3ce0096cd631f0a)
![{\displaystyle =-H_{k}+\sum _{n=1}^{k}\sum _{r=0}^{\omega }{\frac {1}{(n-1)m+m-r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fc6f1c2b503275ed790d0c53adc36b520eaf010)
![{\displaystyle =-H_{k}+\sum _{n=1}^{k}\sum _{j=1}^{m}{\frac {1}{(n-1)m+j}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6855d33e57a534096074ac4fc363a0fa53b64f3)
![{\displaystyle =-H_{k}+\sum _{n=1}^{k}\left(H_{nm}-H_{m(n-1)}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e65e141ea450ddd9451f4625a98724eb47876f5)
![{\displaystyle =-H_{k}+H_{mk}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db6473fac2fbc9496fcbf7b73353a90ddf22508c)
![{\displaystyle \lim _{k\to \infty }=H_{km}-H_{k}=\sum _{x\in \langle m\rangle \cap \mathbb {N} }\sum _{r\in \mathbb {Z} _{m}\cap \mathbb {N} }\left({\frac {1}{x-r}}-{\frac {1}{x}}\right)=\ln(\omega +1)=\ln(m)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad6a63253d8eff4bf7a4e1e558e3e8c51e6472d)
![{\displaystyle \int \ln x\,dx=x\ln x-x+C=x(\ln x-1)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/764f1381e37d05fadaa444e4cea58e7d94dba432)
![{\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C={\frac {x(\ln x-1)}{\ln a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12dcd7530fca6f648a95262051f459a40a63e964)
To remember higher integrals, it is convenient to define
![{\displaystyle x^{\left[n\right]}=x^{n}(\log(x)-H_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be26aeae0c7b88d50e760d2ce40df2af4c44b0bb)
where
is the nth harmonic number:
![{\displaystyle x^{\left[0\right]}=\log x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/078fe3653cf35a30aea1b7f03ea554ae7670b967)
![{\displaystyle x^{\left[1\right]}=x\log(x)-x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c12b6bda581e741822ed456b8e7c42955525db0)
![{\displaystyle x^{\left[2\right]}=x^{2}\log(x)-{\begin{matrix}{\frac {3}{2}}\end{matrix}}x^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/966e2b00f916c63e7ccb68fd3da3908597238c66)
![{\displaystyle x^{\left[3\right]}=x^{3}\log(x)-{\begin{matrix}{\frac {11}{6}}\end{matrix}}x^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0f07636bb06f28bbc8fd84ea091e1fb4b6487f1)
Then
![{\displaystyle {\frac {d}{dx}}\,x^{\left[n\right]}=nx^{\left[n-1\right]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ee1e2e7dc8cef7f0c3a355fcf254c7650852a12)
![{\displaystyle \int x^{\left[n\right]}\,dx={\frac {x^{\left[n+1\right]}}{n+1}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92036b7056a0179b00f4d6739640046d2e8553ba)
Approximating large numbers
The identities of logarithms can be used to approximate large numbers. Note that logb(a) + logb(c) = logb(ac), where a, b, and c are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, 232,582,657 −1. To get the base-10 logarithm, we would multiply 32,582,657 by log10(2), getting 9,808,357.09543 = 9,808,357 + 0.09543. We can then get 109,808,357 × 100.09543 ≈ 1.25 × 109,808,357.
Similarly, factorials can be approximated by summing the logarithms of the terms.
Complex logarithm identities
The complex logarithm is the complex number analogue of the logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However, a multivalued function can be defined which satisfies most of the identities. It is usual to consider this as a function defined on a Riemann surface. A single valued version, called the principal value of the logarithm, can be defined which is discontinuous on the negative x axis, and is equal to the multivalued version on a single branch cut.
Definitions
In what follows, a capital first letter is used for the principal value of functions, and the lower case version is used for the multivalued function. The single valued version of definitions and identities is always given first, followed by a separate section for the multiple valued versions.
- ln(r) is the standard natural logarithm of the real number r.
- Arg(z) is the principal value of the arg function; its value is restricted to (−π, π]. It can be computed using Arg(x + iy) = atan2(y, x).
- Log(z) is the principal value of the complex logarithm function and has imaginary part in the range (−π, π].
![{\displaystyle \operatorname {Log} (z)=\ln(|z|)+i\operatorname {Arg} (z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce3b6b5d895dc34cd7fe1a9deb5a5cdc032c5dcb)
![{\displaystyle e^{\operatorname {Log} (z)}=z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e10733f82bae08d2b858745c7394198cc56ebd60)
The multiple valued version of log(z) is a set, but it is easier to write it without braces and using it in formulas follows obvious rules.
- log(z) is the set of complex numbers v which satisfy ev = z
- arg(z) is the set of possible values of the arg function applied to z.
When k is any integer:
![{\displaystyle \log(z)=\ln(|z|)+i\arg(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/560b7ab47d94f4181b83d97274c381c2c5ff9bdd)
![{\displaystyle \log(z)=\operatorname {Log} (z)+2\pi ik}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d82758a97572baaca63dd3269c0d8815121c0c)
![{\displaystyle e^{\log(z)}=z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d014aa3b80452f9b2d705df46141ecd3d6a77054)
Constants
Principal value forms:
![{\displaystyle \operatorname {Log} (1)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d334814ac9f15b6c502dcf24996227be2387eca5)
![{\displaystyle \operatorname {Log} (e)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b7af5c8d5b041f873b0a3830a0764148ec6e993)
Multiple value forms, for any k an integer:
![{\displaystyle \log(1)=0+2\pi ik}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3df846f92d1458404a21d2af0224bb7f5f1b0e)
![{\displaystyle \log(e)=1+2\pi ik}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e104b9073cfdf336a2158dc04629c206841df536)
Summation
Principal value forms:
![{\displaystyle \operatorname {Log} (z_{1})+\operatorname {Log} (z_{2})=\operatorname {Log} (z_{1}z_{2}){\pmod {2\pi i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9add87793ba23c8eeea2a29eba062bd8690df7ea)
[17] ![{\displaystyle \operatorname {Log} (z_{1})-\operatorname {Log} (z_{2})=\operatorname {Log} (z_{1}/z_{2}){\pmod {2\pi i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30befeea1d445922e2b4d6024f581b3d93b707ed)
[17]
Multiple value forms:
![{\displaystyle \log(z_{1})+\log(z_{2})=\log(z_{1}z_{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95bc3196ab347dc2e6af9e9d9e5c51205058ac3b)
![{\displaystyle \log(z_{1})-\log(z_{2})=\log(z_{1}/z_{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51a1fbec8466425b834aa1a26899fe21fb0d0180)
Powers
A complex power of a complex number can have many possible values.
Principal value form:
![{\displaystyle {z_{1}}^{z_{2}}=e^{z_{2}\operatorname {Log} (z_{1})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4647cc88b2a865c49818e02c9a70137186036e5a)
![{\displaystyle \operatorname {Log} {\left({z_{1}}^{z_{2}}\right)}=z_{2}\operatorname {Log} (z_{1}){\pmod {2\pi i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ab92121a1a0880e884ca990b296e5e62513f5c)
Multiple value forms:
![{\displaystyle {z_{1}}^{z_{2}}=e^{z_{2}\log(z_{1})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd73505df93592f86d12d4ffd4e31974850c407)
Where k1, k2 are any integers:
![{\displaystyle \log {\left({z_{1}}^{z_{2}}\right)}=z_{2}\log(z_{1})+2\pi ik_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c9144d397eabeacc01351179bf46b6d8162e331)
![{\displaystyle \log {\left({z_{1}}^{z_{2}}\right)}=z_{2}\operatorname {Log} (z_{1})+z_{2}2\pi ik_{1}+2\pi ik_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca87e9aae0c0c69013e8e60a841a70220cd483d9)
See also
References
- ^ Weisstein, Eric W. "Logarithm". mathworld.wolfram.com. Retrieved 2020-08-29.
- ^ "4.3 - Properties of Logarithms". people.richland.edu. Retrieved 2020-08-29.
- ^ "Properties and Laws of Logarithms". courseware.cemc.uwaterloo.ca/8. Retrieved 2022-04-23.
- ^ "Archived copy" (PDF). Archived from the original (PDF) on 2016-10-20. Retrieved 2016-12-20.
{{cite web}}
: CS1 maint: archived copy as title (link) - ^ http://www.lkozma.net/inequalities_cheat_sheet/ineq.pdf [bare URL PDF]
- ^ http://downloads.hindawi.com/archive/2013/412958.pdf [bare URL PDF]
- ^ Weisstein, Eric W. "Mercator Series". MathWorld--A Wolfram Web Resource. Retrieved 2024-04-24.
- ^ a b Flajolet, Philippe; Sedgewick, Robert (2009). Analytic Combinatorics. Cambridge University Press. p. 389. ISBN 978-0521898065. See page 117, and VI.8 definition of shifted harmonic numbers on page 389
- ^ a b Deveci, Sinan (2022). "On a Double Series Representation of the Natural Logarithm, the Asymptotic Behavior of Hölder Means, and an Elementary Estimate for the Prime Counting Function". arXiv:2211.10751 [math.NT]. See Theorem 5.2. on pages 22 - 23
- ^ Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley. p. 429. ISBN 0-201-55802-5.
- ^ "Harmonic Number". Wolfram MathWorld. Retrieved 2024-04-24. See formula 13.
- ^ Kifowit, Steven J. (2019). More Proofs of Divergence of the Harmonic Series (PDF) (Report). Prairie State College. Retrieved 2024-04-24. See Proofs 23 and 24 for details on the relationship between harmonic numbers and logarithmic functions.
- ^ Bell, Jordan; Blåsjö, Viktor (2018). "Pietro Mengoli's 1650 Proof That the Harmonic Series Diverges". Mathematics Magazine. 91 (5): 341–347. doi:10.1080/0025570X.2018.1506656. hdl:1874/407528. JSTOR 48665556. Retrieved 2024-04-24.
- ^ Harremoës, Peter (2011). "Is Zero a Natural Number?". arXiv:1102.0418 [math.HO]. A synopsis on the nature of 0 which frames the choice of minimum as the dichotomy between ordinals and cardinals.
- ^ Barton, N. (2020). "Absence perception and the philosophy of zero". Synthese. 197 (9): 3823–3850. doi:10.1007/s11229-019-02220-x. PMC 7437648. PMID 32848285. See section 3.1
- ^ The
shift is characteristic of the right Riemann sum employed to prevent the integral from degenerating into the harmonic series, thereby averting divergence. Here,
functions analogously, serving to regulate the series. The successor operation
signals the implicit inclusion of the modulus
(the region omitted from
). The importance of this, from an axiomatic perspective, becomes evident when the residues of
are formulated as
, where
is bootstrapped by
to produce the residues of modulus
. Consequently,
represents a limiting value in this context. - ^ a b Abramowitz, Milton (1965). Handbook of mathematical functions, with formulas, graphs, and mathematical tables. Irene A. Stegun. New York: Dover Publications. ISBN 0-486-61272-4. OCLC 429082.
External links
A lesson on logarithms can be found on Wikiversity - Logarithm in Mathwords