U matematici postoje nekoliko logaritamskih jednačina.
Algebarske jednačine
Korišćenje jednostavnijih operacija
Ljudi koriste logaritme da bi uprostili račun. Na primer, dva broja mogu biti pomnožena samo koristeći tablicu logaritama i sabiranje.
![{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y)\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c4357f1cdd9ab6b88f9a51c4d18e8c5197ddc1b) | zbog | |
![{\displaystyle \log _{b}\!\left({\begin{matrix}{\frac {x}{y}}\end{matrix}}\right)=\log _{b}(x)-\log _{b}(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c5de40724bd183844957d4c17c7812831006b7c) | zbog | |
![{\displaystyle \log _{b}(x^{y})=y\log _{b}(x)\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6485980ba2ed99b52c491c5e5cbbb19f3e4688) | zbog | |
![{\displaystyle \log _{b}\!\left(\!{\sqrt[{y}]{x}}\right)={\begin{matrix}{\frac {\log _{b}(x)}{y}}\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa4327beb2984c4f0548bee011606d7588db706) | zbog | |
Ukidanje eksponenata
Logaritmi i eksponenti (antilogaritmi) sa istom osnovom se poništavaju.
![{\displaystyle b^{\log _{b}(x)}=x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/279efc9f676f2d56705091a6a06484d0ed2e05db) | zbog | |
![{\displaystyle \log _{b}(b^{x})=x\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c621245b883b70f51ff84cd7dcbf71f42717743) | zbog | |
Promena osnove
![{\displaystyle \log _{a}b={\log _{c}b \over \log _{c}a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14fb3ff79634a5303ca921f4f9bf2cd6fbae1b43)
Ova jednačina se koristi za izračunavanje logaritama na elektronskim kalkulatorima. Na primer, većina kalkulatora ima dugmad za ln i za log10, ali ne i za log2. Da bismo našli log2(3), treba izračunati log10(3) / log10(2) (ili ln(3)/ln(2), što je zapravo ista stvar).
Iz ove formule proizilazi nekoliko stvari:
![{\displaystyle \log _{a}b={\frac {1}{\log _{b}a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15dbdb202a6378003932e79acd2bb1445ab79b78)
![{\displaystyle \log _{a^{n}}b={\frac {1}{n}}\log _{a}b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad482d75c3c9c44a4817ac3809be2082dba89945)
![{\displaystyle a^{\log _{b}c}=c^{\log _{b}a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16f5410f566d0437fa4e94f46fe96659a4676013)
Trivijalne jednačine
![{\displaystyle \log _{b}(1)=0\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d722057c172d6205d8fa4f9639ba59195dcd4fa) | zbog | |
![{\displaystyle \log _{b}(b)=1\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60ea1710fc1dd65eb503ff34f4e60ed931867574) | zbog | |
Jednačine matematičke analize
Limesi
![{\displaystyle \lim _{x\to 0^{+}}\log _{a}x=-\infty \quad {\mbox{if }}a>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6178e257b73c07ffa1eb055619baeee9c608f98f)
![{\displaystyle \lim _{x\to 0^{+}}\log _{a}x=\infty \quad {\mbox{if }}a<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e25e3db784008200f071c9b1accfe4d70c838a4e)
![{\displaystyle \lim _{x\to \infty }\log _{a}x=\infty \quad {\mbox{if }}a>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee3149844beade9b842523133a4e16eb3333fdd7)
![{\displaystyle \lim _{x\to \infty }\log _{a}x=-\infty \quad {\mbox{if }}a<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d00d057aa3b747a9db38732dac7233f3e3d27a12)
![{\displaystyle \lim _{x\to 0^{+}}x^{b}\log _{a}x=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba195e3df473541352d62701ffdc4df98b6bd816)
![{\displaystyle \lim _{x\to \infty }{1 \over x^{b}}\log _{a}x=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41894b8cf122d7336bcda5352973c934faef04e0)
Poslednji limes se često shvata kao "logaritam raste sporije od bilo kog stepena ili korena x".
Izvod logaritamske funkcije
![{\displaystyle {d \over dx}\ln x={1 \over x}={1 \over x\ln e},\qquad x>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/666198eb8843a6ac91840e2987f806463892ed82)
![{\displaystyle {d \over dx}\log _{b}x={1 \over x\ln b},\qquad b>0,b\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf4e4d2a50946aa9d4c1a67072f0b24f5bc7249)
Integral logaritamske funkcije
![{\displaystyle \int \log _{a}x\,dx=x(\log _{a}x-\log _{a}e)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e9d4b89241e2696ab222e6e33cb73c928a62af)
što se za a=e svodi na
![{\displaystyle \int \ln x\,dx=x(\ln x-1)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efc8f92df9765debfdccfb6c701d7bd2d312fb55)