# 6.2: Exponential and Logarithmic Functions

**At Grade**Created by: CK-12

## Learning Objectives

A student will be able to:

- Understand and use the basic definitions of exponential and logarithmic functions and how they are related algebraically.

- Distinguish between an exponential and logarithmic functions graphically.

## A Quick Algebraic Review of Exponential and Logarithmic Functions

**Exponential Functions**

Recall from algebra that an exponential function is a function that has a constant base and a variable exponent. A function of the form \begin{align*}f(x) = b^x\end{align*}

**Logarithmic Functions**

Recall from your previous courses in algebra that a logarithm is an exponent. If the base \begin{align*}b > 0\end{align*}

\begin{align*}y = \log_b x.\end{align*}

This is equivalent to the exponential form

\begin{align*}b^y = x.\end{align*}

For example, the following table shows the logarithmic forms in the first row and the corresponding exponential forms in the second row.

\begin{align*}& \text{Logarithmic Form} \rightarrow & & \log_2 16 = 4 & & \log_5 \frac{1} {25} = -2 & & \log_{10} 100 = 2 & & \log_e e = 1\\ & \text{Exponential Form} \rightarrow & & 2^4 = 16 & & 5^{-2} = \frac{1}{25} & & 10^2 = 100 & & e^1 = e\end{align*}

Historically, logarithms with base of \begin{align*}10\end{align*}

\begin{align*}\lim_{x \rightarrow 0} (1 + x)^{1/x} = e.\end{align*}

We denote the natural logarithm of \begin{align*}x\end{align*} by \begin{align*}\ln x\end{align*} rather than \begin{align*}\log_e x.\end{align*} So keep in mind, that \begin{align*}\ln x\end{align*} is the power to which \begin{align*}e\end{align*} must be raised to produce \begin{align*}x.\end{align*} That is, the following two expressions are equivalent:

\begin{align*} y = \ln x \Longleftrightarrow x = e^y\end{align*}

The table below shows this operation.

\begin{align*}& \text{Natural Logarithm} \ \ln & & \ln 2 = 0.693 & & \ln 1 = 0 & & \ln e = 1 & & \ln e^3 = 3\\ & \text{Equivalent Exponential Form} & & e^{0.693} = 2 & & e^0 = 1 & & e^1 = e & & e^3 = e^3\end{align*}

**A Comparison between Logarithmic Functions and Exponential Functions**

Looking at the two graphs of exponential functions above, we notice that both pass the horizontal line test. This means that an exponential function is a one-to-one function and thus has an inverse. To find a formula for this inverse, we start with the exponential function

\begin{align*}y = b^x.\end{align*}

Interchanging \begin{align*}x\end{align*} and \begin{align*}y,\end{align*}

\begin{align*}x = b^y.\end{align*}

Projecting the logarithm to the base \begin{align*}b\end{align*} on both sides,

\begin{align*}\log_b x & = \log_b b^y\\ & = y\log_b b\\ & = y(1)\\ & = y.\end{align*}

Thus \begin{align*}y = f^{-1}(x) = \log_b x\end{align*} is the inverse of \begin{align*}y = f(x) = b^x.\end{align*}

This implies that the graphs of \begin{align*}f\end{align*} and \begin{align*}f^{-1}\end{align*} are reflections of one another about the line \begin{align*}y = x.\end{align*} The figure below shows this relationship.

Similarly, in the special case when the base \begin{align*}b = e,\end{align*} the two equations above take the forms

\begin{align*}y = f(x) = e^x\end{align*}

and

\begin{align*}f^{-1}(x) = \ln x.\end{align*}

The graph below shows this relationship:

Before we move to the calculus of exponential and logarithmic functions, here is a summary of the two important relationships that we have just discussed:

- The function \begin{align*}y = b^x\end{align*} is equivalent to \begin{align*}x = \log_b y\end{align*} if \begin{align*}y > 0\end{align*} and \begin{align*}x \in R.\end{align*}
- The function \begin{align*}y = e^x\end{align*} is equivalent to \begin{align*}x = \ln y\end{align*} if \begin{align*}y > 0\end{align*} and \begin{align*}x \in R.\end{align*}

You should also recall the following important properties about logarithms:

- \begin{align*} \log_b {vw} = \log_b{v} + \log_b {w}\end{align*}
- \begin{align*} \log_b {\frac {v}{w}} = \log_b {v} - \log_b {w}\end{align*}
- \begin{align*} \log_b {w^n} = n \log_b {w}\end{align*}
- To express a logarithm with base in terms of the natural logarithm: \begin{align*} \log_b {w} = {\frac {\ln w} {\ln b}} \end{align*}
- To express a logarithm with base \begin{align*} b\end{align*} in terms of another base \begin{align*}a\end{align*}: \begin{align*}\log_b {w} = \frac {\log_a{w}} {\log_a{b}}\end{align*}

## Review Questions

Solve for \begin{align*}x.\end{align*}

- \begin{align*} 6^x = \frac {1}{216}\end{align*}
- \begin{align*}e^x = 3\end{align*}
- \begin{align*}\log_2 {z} = 3\end{align*}
- \begin{align*}\ln {x^2} = 5\end{align*}
- \begin{align*}3e^{-5x} = 132\end{align*}
- \begin{align*}e^{2x}- 7e^x + 10 = 0\end{align*}
- \begin{align*}-4(3)^x = -36\end{align*}
- \begin{align*}\ln x - \ln 3 = 2\end{align*}
- \begin{align*} y = 5 \log_{10} \left({\frac {2} {2-x}}\right)\end{align*}
- \begin{align*}y = 3e^{-2x/3}\end{align*}

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## Date Created:

Feb 23, 2012## Last Modified:

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