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6.2: Exponential and Logarithmic Functions

Difficulty Level: 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*} where \begin{align*}b\end{align*} is a constant and \begin{align*}b > 0\end{align*} and \begin{align*}b \neq 1\end{align*} is called an exponential function with base \begin{align*}b.\end{align*} Some examples are \begin{align*}f(x) = 2^x,\end{align*} \begin{align*} f(x) = \left ( \frac{1} {2}\right )^x,\end{align*} and \begin{align*}f(x) = e^x.\end{align*} All exponential functions are continuous and their graph is one of the two basic shapes, depending on whether \begin{align*}0 < b < 1\end{align*} or \begin{align*}b > 1.\end{align*} The graph below shows the two basic shapes:

Logarithmic Functions

Recall from your previous courses in algebra that a logarithm is an exponent. If the base \begin{align*}b > 0\end{align*} and \begin{align*}b \neq 1,\end{align*} then for any value of \begin{align*}x > 0,\end{align*} the logarithm to the base \begin{align*}b\end{align*} of the value of \begin{align*}x\end{align*} is denoted by

\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*} were very popular. They are called the common logarithms. Recently the base \begin{align*}2\end{align*} has been gaining popularity due to its considerable role in the field of computer science and the associated binary number system. However, the most widely used base in applications is the natural logarithm, which has an irrational base denoted by \begin{align*}e,\end{align*} in honor of the famous mathematician Leonhard Euler. This irrational constant is \begin{align*}e \approx 2.718281.\end{align*} Formally, it is defined as the limit of \begin{align*}(1 + x)^{1/x}\end{align*} as \begin{align*}x\end{align*} approaches zero. That is,

\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*}

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

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