Definition of the logarithm

The logarithm $\log_a b$ answers one question: $a$ to what power gives $b$? The definition is $\log_a b=c \iff a^c=b$, so $\log_2 8=3$ is just $2^3=8$ rewritten.

Translating the exponent rules into logarithms gives three rules.

In calculus, $\log x$ means the natural logarithm (base $e$). The rule $(\log x)'=\frac{1}{x}$ follows from this definition and the derivative of $e^x$.

Examples

1. 01Given $\log_{5}125 = 3$, evaluate:$\displaystyle \log_{5}625 - \log_{5}125$AnswerHide

   Answer$\displaystyle 1$
2. 02Given $\log_{2}4 = 2$, evaluate:$\displaystyle \log_{2}2 + \log_{2}4$AnswerHide

   Answer$\displaystyle 3$
3. 03Write in exponential form:$\displaystyle \log_{2}8 = 3$AnswerHide

   Answer$\displaystyle 2^{3} = 8$
4. 04Given $\log_{2}4 = 2$, evaluate:$\displaystyle \log_{2}8 - \log_{2}4$AnswerHide

   Answer$\displaystyle 1$
5. 05Given $\log_{2}8 = 3$, evaluate:$\displaystyle \log_{2}4 + \log_{2}8$AnswerHide

   Answer$\displaystyle 5$
6. 06Given $\log_{2}2 = 1$, evaluate:$\displaystyle \log_{2}4 + \log_{2}2$AnswerHide

   Answer$\displaystyle 3$

FAQ

Q1What is the base of log x?Show answerHide

A

In this app and in calculus, $\log x$ with no base means the natural logarithm (base $e$). The common logarithm is written $\log_{10} x$.

Practice this pattern in the app, 5 problems a day.

Published 2026-09-04 · Updated 2026-09-05