Logarithm Properties Reference Cheat Sheet

Key Facts

• The logarithmic equation $\log_b(x)=y$ is equivalent to the exponential equation $b^y=x$, where $b>0$, $b\ne 1$, and $x>0$.
• The product property is $\log_b(MN)=\log_b(M)+\log_b(N)$ for $M>0$ and $N>0$.
• The quotient property is $\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$ for $M>0$ and $N>0$.
• The power property is $\log_b(M^p)=p\log_b(M)$ for $M>0$.
• The change of base formula is $\log_b(M)=\frac{\log_a(M)}{\log_a(b)}$, where $a>0$, $a\ne 1$, $b>0$, and $b\ne 1$.
• The inverse properties are $\log_b(b^x)=x$ and $b^{\log_b(x)}=x$ for $x>0$.
• The special values are $\log_b(1)=0$ and $\log_b(b)=1$ for $b>0$ and $b\ne 1$.
• A logarithm with base $10$ is written $\log(x)$, and a logarithm with base $e$ is written $\ln(x)$.

Vocabulary

Logarithm

A logarithm $\log_b(x)$ is the exponent that base $b$ must be raised to in order to get $x$.

Base

The base $b$ in $\log_b(x)$ is the number being raised to a power, and it must satisfy $b>0$ and $b\ne 1$.

Argument

The argument is the input $x$ in $\log_b(x)$, and it must be positive.

Common Logarithm

A common logarithm is a logarithm with base $10$, written as $\log(x)$.

Natural Logarithm

A natural logarithm is a logarithm with base $e$, written as $\ln(x)$.

Change of Base

Change of base rewrites $\log_b(M)$ as $\frac{\log_a(M)}{\log_a(b)}$ so it can be evaluated using another base.