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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*}


\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.


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