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## Subsection2.2Logarithmic Functions

Recall inverse functions. If for each $y$ in the range of $f$ there exists exactly one value of $x$ such that $f(x)=y,$ then $f$ has an inverse at $y$ denoted by $f^{-1}$ such that

Hence

The inverse of the exponential function $f(x)=e^x$ is the natural logarithmic function $f^{-1}(x)= \ln(x)$ so we have

\begin{equation*} \ln(x) =c \iff e^c=x \end{equation*}
###### The Logarithm

Let $b\neq 1$ be a positive number, then the function

\begin{equation*} f(t)=\log_b(t) \end{equation*}

is called a logarithm with base $b \text{.}$

Upon inputting a value $t\text{,}$ the function $\log_b(t)$ will tell you the power of $b$ which will yield $t \text{.}$

Due to the relationship between logarithms and exponentials, we often say that the equations

\begin{equation*} x=\log_b(y)\ \ \ \ \text{and}\ \ \ \ b^x=y \end{equation*}

are equivalent.

###### Warning2.28

Two common bases of a logarithm are base $10$ and $e\text{.}$ Since they are used so often, we have developed a short hand notation for a logarithm of base $10$ and a logarithm of base $e\text{.}$ This short hand is shown below:

\begin{equation*} \log_{10}(y)=\log(y). \end{equation*}
\begin{equation*} \log_{e}(y)=\ln(y). \end{equation*}

In other words, we simply drop the subscript when referring to base $10$ and we change to $\ln$ when referring to base $e$

There are some important properties of logarithms that you should be familiar with.

###### Properties of Logarithms

If $x,y,b>0\text{,}$ and $b\neq 1\text{,}$ then

1. $\log_b(xy)=\log_b(x)+\log_b(y),$

2. $\log_b(\frac{x}{y})=\log_b(x)-\log_b(y),$

3. $\log_b(x^k)=k\cdot \log_b(x),$

4. $\log_b(b^y)=y,$

5. $b^{\log_b(x)}=x.$