# 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

**Logarithmic Functions**

Recall from your previous courses in algebra that a logarithm is an exponent. If the base

This is equivalent to the exponential form

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

Historically, logarithms with base of

We denote the natural logarithm of

The table below shows this operation.

**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

Interchanging

Projecting the logarithm to the base

Thus

This implies that the graphs of

Similarly, in the special case when the base

and

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
y=bx is equivalent tox=logby ify>0 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*}

## Review Answers

- \begin{align*}x = -3\end{align*}
- \begin{align*}\ln {3}\end{align*}
- \begin{align*}z = 8\end{align*}
- \begin{align*}e^{5/2}\end{align*}
- \begin{align*}x = -0.757\end{align*}
- \begin{align*}\ln 2\end{align*} and \begin{align*}\ln 5\end{align*}
- \begin{align*}x = 2\end{align*}
- \begin{align*}x = 3e^2\end{align*}
- \begin{align*} x = 2\left (1 - \frac {1} {10^{y/5}}\right )\end{align*}
- \begin{align*}x = - \frac {3}{2} \ln \left (\frac {y}{3}\right )\end{align*}

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