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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

\begin{equation*}
f(x)=y \quad \iff \quad f^{-1}(y)=x
\end{equation*}

Hence

\begin{equation*}
f(f^{-1}(y))=y \quad \text{and} \quad f^{-1}(f(x))=x.
\end{equation*}

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.

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

\(\log_b(xy)=\log_b(x)+\log_b(y),\)

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

\(\log_b(x^k)=k\cdot \log_b(x),\)

\(\log_b(b^y)=y,\)

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