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
\(\log_b(xy)=\log_b(x)+\log_b(y),\)
\(\log_b(\frac{x}{y})=\log_b(x)-\log_b(y),\)
\(\log_b(x^k)=k\cdot \log_b(x),\)
\(\log_b(b^y)=y,\)
\(b^{\log_b(x)}=x.\)