Convert Between Exponential and Logarithmic Form

Since the equations $y={\text{log}}_{a}x$ and $x=a^y$ are equivalent, we can go back and forth between them. This will often be the method to solve some exponential and logarithmic equations. To help with converting back and forth let’s take a close look at the equations. See Figure 10.3. Notice the positions of the exponent and base.

Figure 10.3

If we realize the logarithm is the exponent it makes the conversion easier. You may want to repeat, “base to the exponent give us the number.”

Example 10.18

Convert to logarithmic form: ⓐ $2^3=8,$ ⓑ $5^{\frac{1}{2}}=\sqrt{5},$ and ⓒ ${\left(\frac{1}{2}\right)}^4=\frac{1}{16}.$

Solution

In the next example we do the reverse—convert logarithmic form to exponential form.

Example 10.19

Convert to exponential form: ⓐ $2={\text{log}}_{8}64,$ ⓑ $0={\text{log}}_{4}1,$ and ⓒ $-3={\text{log}}_{10}\frac{1}{1000}.$

Solution

Evaluate Logarithmic Functions

We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential equation.

When see an expression such as ${\text{log}}_{3}27,$ we can find its exact value two ways. By inspection we realize it means $“3$ to what power will be $27”?$ Since $3^3=27,$ we know ${\text{log}}_{3}27=3.$ An alternate way is to set the expression equal to $x$ and then convert it into an exponential equation.

Graph Logarithmic Functions

To graph a logarithmic function $y={\text{log}}_{a}x,$ it is easiest to convert the equation to its exponential form, $x=a^y.$ Generally, when we look for ordered pairs for the graph of a function, we usually choose an x-value and then determine its corresponding y-value. In this case you may find it easier to choose y-values and then determine its corresponding x-value.

Example 10.22

Graph $y={\text{log}}_{2}x.$

Solution

To graph the function, we will first rewrite the logarithmic equation, $y={\text{log}}_{2}x,$ in exponential form, $2^y=x.$

We will use point plotting to graph the function. It will be easier to start with values of y and then get x.

$y$	$2^y=x$	$\left(x,y\right)$
$\mathrm{-2}$	$2^{\mathrm{-2}}=\frac{1}{2^2}=\frac{1}{4}$	$\left(\frac{1}{4},-2\right)$
$\mathrm{-1}$	$2^{\mathrm{-1}}=\frac{1}{2^1}=\frac{1}{2}$	$\left(\frac{1}{2},\mathrm{-1}\right)$
0	$2^0=1$	$\left(1,0\right)$
1	$2^1=2$	$\left(2,1\right)$
2	$2^2=4$	$\left(4,2\right)$
3	$2^3=8$	$\left(8,3\right)$

The graphs of $y={\text{log}}_{2}x,$$y={\text{log}}_{3}x,$ and $y={\text{log}}_{5}x$ are the shape we expect from a logarithmic function where $a>1.$

We notice that for each function the graph contains the point $\left(1,0\right).$ This make sense because $0={\text{log}}_{a}1$ means $a^0=1$ which is true for any a.

The graph of each function, also contains the point $\left(a,1\right).$ This makes sense as $1={\text{log}}_{a}a$ means $a^1=a.$ which is true for any a.

Notice too, the graph of each function $y={\text{log}}_{a}x$ also contains the point $\left(\frac{1}{a},\mathrm{-1}\right).$ This makes sense as $\mathrm{-1}={\text{log}}_a\frac{1}{a}$ means $a^{\mathrm{-1}}=\frac{1}{a},$ which is true for any a.

Look at each graph again. Now we will see that many characteristics of the logarithm function are simply ’mirror images’ of the characteristics of the corresponding exponential function.

What is the domain of the function? The graph never hits the y-axis. The domain is all positive numbers. We write the domain in interval notation as $\left(0,\infty \right).$

What is the range for each function? From the graphs we can see that the range is the set of all real numbers. There is no restriction on the range. We write the range in interval notation as $\left(\text{−}\infty ,\infty \right).$

When the graph approaches the y-axis so very closely but will never cross it, we call the line $x=0,$ the y-axis, a vertical asymptote.

Properties of the Graph of $y={\text{log}}_{a}x$ when $a>1$

Domain	$\left(0,\infty \right)$
Range	$\left(\text{−}\infty ,\infty \right)$
$x\text{-}\text{intercept}$	$\left(1,0\right)$
$y\text{-}\text{intercept}$	None
Contains	$\left(a,1\right),$$\left(\frac{1}{a},\mathrm{-1}\right)$
Asymptote	$y\text{-}\text{axis}$

Our next example looks at the graph of $y={\text{log}}_{a}x$ when $0<a<1.$

Example 10.23

Graph $y={\text{log}}_{\frac{1}{3}}x.$

Solution

To graph the function, we will first rewrite the logarithmic equation, $y={\text{log}}_{\frac{1}{3}}x,$ in exponential form, ${\left(\frac{1}{3}\right)}^y=x.$

We will use point plotting to graph the function. It will be easier to start with values of y and then get x.

$y$	${\left(\frac{1}{3}\right)}^y=x$	$\left(x,y\right)$
$\mathrm{-2}$	${\left(\frac{1}{3}\right)}^{\mathrm{-2}}=3^2=9$	$\left(9,\mathrm{-2}\right)$
$\mathrm{-1}$	${\left(\frac{1}{3}\right)}^{\mathrm{-1}}=3^1=3$	$\left(3,\mathrm{-1}\right)$
0	${\left(\frac{1}{3}\right)}^0=1$	$\left(1,0\right)$
1	${\left(\frac{1}{3}\right)}^1=\frac{1}{3}$	$\left(\frac{1}{3},1\right)$
2	${\left(\frac{1}{3}\right)}^2=\frac{1}{9}$	$\left(\frac{1}{9},2\right)$
3	${\left(\frac{1}{3}\right)}^3=\frac{1}{27}$	$\left(\frac{1}{27},3\right)$

Now, let’s look at the graphs $y={\text{log}}_{\frac{1}{2}}x,y={\text{log}}_{\frac{1}{3}}x$ and $y={\text{log}}_{\frac{1}{4}}x$, so we can identify some of the properties of logarithmic functions where $0<a<1.$

The graphs of all have the same basic shape. While this is the shape we expect from a logarithmic function where $0<a<1.$

We notice, that for each function again, the graph contains the points,$\left(1,0\right),$$\left(a,1\right),$$\left(\frac{1}{a},\mathrm{-1}\right).$ This make sense for the same reasons we argued above.

We notice the domain and range are also the same—the domain is $\left(0,\infty \right)$ and the range is $\left(\text{−}\infty ,\infty \right).$ The $y$-axis is again the vertical asymptote.

We will summarize these properties in the chart below. Which also include when $a>1.$

Properties of the Graph of $y={\text{log}}_{a}x$

when $a>1$		when $0<a<1$
Domain	$\left(0,\infty \right)$	Domain	$\left(0,\infty \right)$
Range	$\left(\text{−}\infty ,\infty \right)$	Range	$\left(\text{−}\infty ,\infty \right)$
$x$-intercept	$\left(1,0\right)$	$x$-intercept	$\left(1,0\right)$
$y$-intercept	none	$y$-intercept	None
Contains	$\left(a,1\right),$$\left(\frac{1}{a},\mathrm{-1}\right)$	Contains	$\left(a,1\right),$$\left(\frac{1}{a},\mathrm{-1}\right)$
Asymptote	$y$-axis	Asymptote	$y$-axis
Basic shape	increasing	Basic shape	Decreasing

We talked earlier about how the logarithmic function $f^{\mathrm{-1}}\left(x\right)={\text{log}}_{a}x$ is the inverse of the exponential function $f\left(x\right)=a^x.$ The graphs in Figure 10.4 show both the exponential (blue) and logarithmic (red) functions on the same graph for both $a>1$ and $0<a<1.$

Figure 10.4

Notice how the graphs are reflections of each other through the line $y=x.$ We know this is true of inverse functions. Keeping a visual in your mind of these graphs will help you remember the domain and range of each function. Notice the x-axis is the horizontal asymptote for the exponential functions and the y-axis is the vertical asymptote for the logarithmic functions.

Solve Logarithmic Equations

When we talked about exponential functions, we introduced the number e. Just as e was a base for an exponential function, it can be used a base for logarithmic functions too. The logarithmic function with base e is called the natural logarithmic function. The function $f\left(x\right)={\text{log}}_{e}x$ is generally written $f\left(x\right)=\text{ln}x$ and we read it as “el en of $x.”$

When the base of the logarithm function is 10, we call it the common logarithmic function and the base is not shown. If the base a of a logarithm is not shown, we assume it is 10.

To solve logarithmic equations, one strategy is to change the equation to exponential form and then solve the exponential equation as we did before. As we solve logarithmic equations, $y={\text{log}}_{a}x$, we need to remember that for the base a, $a>0$ and $a\ne 1.$ Also, the domain is $x>0.$ Just as with radical equations, we must check our solutions to eliminate any extraneous solutions.

Use Logarithmic Models in Applications

There are many applications that are modeled by logarithmic equations. We will first look at the logarithmic equation that gives the decibel (dB) level of sound. Decibels range from 0, which is barely audible to 160, which can rupture an eardrum. The ${10}^{\mathrm{-12}}$ in the formula represents the intensity of sound that is barely audible.

Example 10.26

Extended exposure to noise that measures 85 dB can cause permanent damage to the inner ear which will result in hearing loss. What is the decibel level of music coming through ear phones with intensity ${10}^{\mathrm{-2}}$ watts per square inch?

Solution

Substitute in the intensity level, I.
Simplify.
Since $\text{log}{10}^{10}=10.$
Multiply.
	The decibel level of music coming through earphones is 100 dB.

The magnitude $R$ of an earthquake is measured by a logarithmic scale called the Richter scale. The model is $R=\text{log}I,$ where $I$ is the intensity of the shock wave. This model provides a way to measure earthquake intensity.