Evaluate and Graph Logarithmic Functions

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 . Notice the positions of the exponent and base.

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

Try it.

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

Solution

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

Example

Try it.

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.

Example

Try it.

Find the value of x: ⓐ \({\text{log}}_{x}36=2,\) ⓑ \({\text{log}}_{4}x=3,\) and ⓒ \({\text{log}}_{\frac{1}{2}}\frac{1}{8}=x.\)

Solution

ⓐ

	\(\ {\text{log}}_{x}36\ =\ 2\)
Convert to exponential form.	\(\ {x}^{2}\ =\ 36\)
Solve the quadratic.	\(\ x=6,\ x=-6\)
The base of a logarithmic function must be positive, so we eliminate \(x=-6\).	\(\ x\ =\ 6\ \text{Therefore,}\ {\text{log}}_{6}36=2.\)

ⓑ

	\(\ {\text{log}}_{4}x\ =\ 3\)
Convert to exponential form.	\(\ {4}^{3}\ =\ x\)
Simplify.	\(\ x\ =\ 64\ \text{Therefore,}\ {\text{log}}_{4}64\ =\ 3.\)

ⓒ

	\({\text{log}}_{\frac{1}{2}}\frac{1}{8}\ =\ x\)
Convert to exponential form.	\(\ {(\frac{1}{2})}^{x}\ =\ \frac{1}{8}\)
Rewrite \(\frac{1}{8}\) as \({(\frac{1}{2})}^{3}\).	\(\ {(\frac{1}{2})}^{x}\ =\ {(\frac{1}{2})}^{3}\)
With the same base, the exponents must be equal.	\(\ x\ =\ 3\ \text{Therefore,}\ {\text{log}}_{\frac{1}{2}}\frac{1}{8}=3\)

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.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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

Try it.

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.

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

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 \((1,0).\) 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 \((a,1).\) 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 \((\frac{1}{a},-1).\) This makes sense as \(-1={\text{log}}_{a}\frac{1}{a}\) means \({a}^{-1}=\frac{1}{a},\) 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 \((0,\infty ).\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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 \(f(x)={\text{log}}_{e}x\) is generally written \(f(x)=\text{ln}\ x\) 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.

Example

Try it.

Solve: ⓐ \({\text{log}}_{a}49=2\) and ⓑ \(\text{ln}\ x=3.\)

Solution

ⓐ

	\({\log }_{a}49\ =\ 2\)
Rewrite in exponential form.	\(\ {a}^{2}\ =\ 49\)
Solve the equation using the square root property.	\(\ a\ =\ \pm 7\)
The base cannot be negative, so we eliminate \(a=-7.\)	\(a=7,\ a=-7\)
Check.
\(\begin{array}{llll}a=7 & {\log }_{a}49 & = & 2 \\ & {\log }_{7}49 & \overset{?}{=} & 2 \\ & {7}^{2} & \overset{?}{=} & 49 \\ & 49 & = & 49✓\end{array}\)

ⓑ

	\(\ \text{ln}\ x\ =\ 3\)
Rewrite in exponential form.	\(\ {e}^{3}\ =\ x\)
Check.
\(\begin{array}{llll} \\ x={e}^{3} & \text{ln}\ x & = & 3 \\ & \text{ln}\ {e}^{3} & \overset{?}{=} & 3 \\ & {e}^{3} & = & {e}^{3}✓\end{array}\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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 \({10}^{-12}\) in the formula represents the intensity of sound that is barely audible.

Example

Try it.

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 \({10}^{-2}\) 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.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

• Properties of the Graph of \(y={\text{log}}_{a}x:\)
  

  when \(a>1\)	when \(0
  Domain	\((0,\infty )\)	Domain	\((0,\infty )\)
  Range	\((\text{-}\infty ,\infty )\)	Range	\((\text{-}\infty ,\infty )\)
  x-intercept	\((1,0)\)	x-intercept	\((1,0)\)
  y-intercept	none	y-intercept	none
  Contains	\((a,1),\)\((\frac{1}{a},-1)\)	Contains	\((a,1),\)\((\frac{1}{a},-1)\)
  Asymptote	y-axis	Asymptote	y-axis
  Basic shape	increasing	Basic shape	decreasing

  
  
• Decibel Level of Sound: The loudness level, \(D\), measured in decibels, of a sound of intensity, \(I\), measured in watts per square inch is \(D=10\text{log}(\frac{I}{{10}^{-12}}).\)
• Earthquake Intensity: The magnitude \(R\) of an earthquake is measured by \(R=\text{log}\ I,\) where \(I\) is the intensity of its shock wave.

Evaluate and Graph Logarithmic Functions

Convert Between Exponential and Logarithmic Form

In the following exercises, convert from exponential to logarithmic form.

Try it.

\({4}^{2}=16\)

Try it.

\({2}^{5}=32\)

Solution

\({\text{log}}_{2}32=5\)

Try it.

\({3}^{3}=27\)

Try it.

\({5}^{3}=125\)

Solution

\({\text{log}}_{5}125=3\)

Try it.

\({10}^{3}=1000\)

Try it.

\({10}^{-2}=\frac{1}{100}\)

Solution

\(\text{log}\frac{1}{100}=-2\)

Try it.

\({x}^{\frac{1}{2}}=\sqrt{3}\)

Try it.

\({x}^{\frac{1}{3}}=\sqrt[3]{6}\)

Solution

\({\text{log}}_{x}\sqrt[3]{6}=\frac{1}{3}\)

Try it.

\({32}^{x}=\sqrt[4]{32}\)

Try it.

\({17}^{x}=\sqrt[5]{17}\)

Solution

\({\text{log}}_{17}\sqrt[5]{17}=x\)

Try it.

\({(\frac{1}{4})}^{2}=\frac{1}{16}\)

Try it.

\({(\frac{1}{3})}^{4}=\frac{1}{81}\)

Solution

\({\text{log}}_{\frac{1}{3}}\frac{1}{81}=4\)

Try it.

\({3}^{-2}=\frac{1}{9}\)

Try it.

\({4}^{-3}=\frac{1}{64}\)

Solution

\({\text{log}}_{4}\frac{1}{64}=-3\)

Try it.

\({e}^{x}=6\)

Try it.

\({e}^{3}=x\)

Solution

\(\text{ln}\ x=3\)

In the following exercises, convert each logarithmic equation to exponential form.

Try it.

\(3={\text{log}}_{4}64\)

Try it.

\(6={\text{log}}_{2}64\)

Solution

\(64={2}^{6}\)

Try it.

\(4={\text{log}}_{x}81\)

Try it.

\(5={\text{log}}_{x}32\)

Solution

\(32={x}^{5}\)

Try it.

\(0={\text{log}}_{12}1\)

Try it.

\(0={\text{log}}_{7}1\)

Solution

\(1={7}^{0}\)

Try it.

\(1={\text{log}}_{3}3\)

Try it.

\(1={\text{log}}_{9}9\)

Solution

\(9={9}^{1}\)

Try it.

\(-4={\text{log}}_{10}\frac{1}{10,000}\)

Try it.

\(3={\text{log}}_{10}1,000\)

Solution

\(1,000={10}^{3}\)

Try it.

\(5={\text{log}}_{e}x\)

Try it.

\(x={\text{log}}_{e}43\)

Solution

\(43={e}^{x}\)

Evaluate Logarithmic Functions

In the following exercises, find the value of \(x\) in each logarithmic equation.

Try it.

\({\text{log}}_{x}49=2\)

Try it.

\({\text{log}}_{x}121=2\)

Solution

\(x=11\)

Try it.

\({\text{log}}_{x}27=3\)

Try it.

\({\text{log}}_{x}64=3\)

Solution

\(x=4\)

Try it.

\({\text{log}}_{3}x=4\)

Try it.

\({\text{log}}_{5}x=3\)

Solution

\(x=125\)

Try it.

\({\text{log}}_{2}x=-6\)

Try it.

\({\text{log}}_{3}x=-5\)

Solution

\(x=\frac{1}{243}\)

Try it.

\({\text{log}}_{\frac{1}{4}}\frac{1}{16}=x\)

Try it.

\({\text{log}}_{\frac{1}{3}}\frac{1}{9}=x\)

Solution

\(x=2\)

Try it.

\({\text{log}}_{\frac{1}{4}}64=x\)

Try it.

\({\text{log}}_{\frac{1}{9}}81=x\)

Solution

\(x=-2\)

In the following exercises, find the exact value of each logarithm without using a calculator.

Try it.

\({\text{log}}_{7}49\)

Try it.

\({\text{log}}_{6}36\)

Solution

2

Try it.

\({\text{log}}_{4}1\)

Try it.

\({\text{log}}_{5}1\)

Solution

0

Try it.

\({\text{log}}_{16}4\)

Try it.

\({\text{log}}_{27}3\)

Solution

\(\frac{1}{3}\)

Try it.

\({\text{log}}_{\frac{1}{2}}2\)

Try it.

\({\text{log}}_{\frac{1}{2}}4\)

Solution

\(-2\)

Try it.

\({\text{log}}_{2}\frac{1}{16}\)

Try it.

\({\text{log}}_{3}\frac{1}{27}\)

Solution

\(-3\)

Try it.

\({\text{log}}_{4}\frac{1}{16}\)

Try it.

\({\text{log}}_{9}\frac{1}{81}\)

Solution

\(-2\)

Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

Try it.

\(y={\text{log}}_{2}x\)

Try it.

\(y={\text{log}}_{4}x\)

Solution

Try it.

\(y={\text{log}}_{6}x\)

Try it.

\(y={\text{log}}_{7}x\)

Solution

Try it.

\(y={\text{log}}_{1.5}x\)

Try it.

\(y={\text{log}}_{2.5}x\)

Solution

Try it.

\(y={\text{log}}_{\frac{1}{3}}x\)

Try it.

\(y={\text{log}}_{\frac{1}{5}}x\)

Solution

Try it.

\(y={\text{log}}_{0.4}x\)

Try it.

\(y={\text{log}}_{0.6}x\)

Solution

Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

Try it.

\({\text{log}}_{a}16=2\)

Try it.

\({\text{log}}_{a}81=2\)

Solution

\(a=9\)

Try it.

\({\text{log}}_{a}8=3\)

Try it.

\({\text{log}}_{a}27=3\)

Solution

\(a=3\)

Try it.

\({\text{log}}_{a}32=2\)

Try it.

\({\text{log}}_{a}24=3\)

Solution

\(a=2\sqrt[3]{3}\)

Try it.

\(\text{ln}\ x=5\)

Try it.

\(\text{ln}\ x=4\)

Solution

\(x={e}^{4}\)

Try it.

\({\text{log}}_{2}(5x+1)=4\)

Try it.

\({\text{log}}_{2}(6x+2)=5\)

Solution

\(x=5\)

Try it.

\({\text{log}}_{3}(4x-3)=2\)

Try it.

\({\text{log}}_{3}(5x-4)=4\)

Solution

\(x=17\)

Try it.

\({\text{log}}_{4}(5x+6)=3\)

Try it.

\({\text{log}}_{4}(3x-2)=2\)

Solution

\(x=6\)

Try it.

\(\text{ln}\ {e}^{4x}=8\)

Try it.

\(\text{ln}\ {e}^{2x}=6\)

Solution

\(x=3\)

Try it.

\(\text{log}{x}^{2}=2\)

Try it.

\(\text{log}({x}^{2}-25)=2\)

Solution

\(x=-5\sqrt{5},x=5\sqrt{5}\)

Try it.

\({\text{log}}_{2}({x}^{2}-4)=5\)

Try it.

\({\text{log}}_{3}({x}^{2}+2)=3\)

Solution

\(x=-5,x=5\)

Use Logarithmic Models in Applications

In the following exercises, use a logarithmic model to solve.

Try it.

What is the decibel level of normal conversation with intensity \({10}^{-6}\) watts per square inch?

Try it.

What is the decibel level of a whisper with intensity \({10}^{-10}\) watts per square inch?

Solution

A whisper has a decibel level of 20 dB.

Try it.

What is the decibel level of the noise from a motorcycle with intensity \({10}^{-2}\) watts per square inch?

Try it.

What is the decibel level of the sound of a garbage disposal with intensity \({10}^{-2}\) watts per square inch?

Solution

The sound of a garbage disposal has a decibel level of 100 dB.

Try it.

In 2014, Chile experienced an intense earthquake with a magnitude of \(8.2\) on the Richter scale. In 2010, Haiti also experienced an intense earthquake which measured \(7.0\) on the Richter scale. Compare the intensities of the two earthquakes.

Try it.

The Los Angeles area experiences many earthquakes. In 1994, the Northridge earthquake measured magnitude of \(6.7\) on the Richter scale. In 2014, Los Angeles also experienced an earthquake which measured \(5.1\) on the Richter scale. Compare the intensities of the two earthquakes.

Solution

The intensity of the 1994 Northridge earthquake in the Los Angeles area was about 40 times the intensity of the 2014 earthquake.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Solve: \({x}^{2}=81.\)
   If you missed this problem, review .

   Sýna svarið

   \(x=9,\ x=-9\)

2. Evaluate: \({3}^{-2}.\)
   If you missed this problem, review .

   Sýna svarið

   \(\frac{1}{9}\)

3. Solve: \({2}^{4}=3x-5.\)
   If you missed this problem, review .

   Sýna svarið

   \(x=7\)

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

5. Convert to logarithmic form: ⓐ \({3}^{2}=9\) ⓑ \({7}^{\frac{1}{2}}=\sqrt{7}\) ⓒ \({(\frac{1}{3})}^{x}=\frac{1}{27}\)

   Sýna svarið

   ⓐ \({\text{log}}_{3}9=2\)
   ⓑ \({\text{log}}_{7}\sqrt{7}=\frac{1}{2}\) ⓒ \({\text{log}}_{\frac{1}{3}}\frac{1}{27}=x\)

6. Convert to logarithmic form: ⓐ \({4}^{3}=64\) ⓑ \({4}^{\frac{1}{3}}=\sqrt[3]{4}\) ⓒ \({(\frac{1}{2})}^{x}=\frac{1}{32}\)

   Sýna svarið

   ⓐ \({\text{log}}_{4}64=3\)
   ⓑ \({\text{log}}_{4}\sqrt[3]{4}=\frac{1}{3}\) ⓒ \({\text{log}}_{\frac{1}{2}}\frac{1}{32}=x\)

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

8. Convert to exponential form: ⓐ \(3={\text{log}}_{4}64\) ⓑ \(0={\text{log}}_{x}1\) ⓒ \(-2={\text{log}}_{10}\frac{1}{100}\)

   Sýna svarið

   ⓐ \(64={4}^{3}\)
   ⓑ \(1={x}^{0}\) ⓒ \(\frac{1}{100}={10}^{-2}\)

9. Convert to exponential form: ⓐ \(3={\text{log}}_{3}27\) ⓑ \(0={\text{log}}_{3}1\) ⓒ \(-1={\text{log}}_{10}\frac{1}{10}\)

   Sýna svarið

   ⓐ \(27={3}^{3}\) ⓑ \(1={3}^{0}\)
   ⓒ \(\frac{1}{10}={10}^{-1}\)

10. Find the value of x: ⓐ \({\text{log}}_{x}36=2,\) ⓑ \({\text{log}}_{4}x=3,\) and ⓒ \({\text{log}}_{\frac{1}{2}}\frac{1}{8}=x.\)

    Sýna svarið

    ⓐ
    

    	\(\ {\text{log}}_{x}36\ =\ 2\)
    Convert to exponential form.	\(\ {x}^{2}\ =\ 36\)
    Solve the quadratic.	\(\ x=6,\ x=-6\)
    The base of a logarithmic function must be positive, so we eliminate \(x=-6\).	\(\ x\ =\ 6\ \text{Therefore,}\ {\text{log}}_{6}36=2.\)

    ⓑ
    

    	\(\ {\text{log}}_{4}x\ =\ 3\)
    Convert to exponential form.	\(\ {4}^{3}\ =\ x\)
    Simplify.	\(\ x\ =\ 64\ \text{Therefore,}\ {\text{log}}_{4}64\ =\ 3.\)

    ⓒ
    

    	\({\text{log}}_{\frac{1}{2}}\frac{1}{8}\ =\ x\)
    Convert to exponential form.	\(\ {(\frac{1}{2})}^{x}\ =\ \frac{1}{8}\)
    Rewrite \(\frac{1}{8}\) as \({(\frac{1}{2})}^{3}\).	\(\ {(\frac{1}{2})}^{x}\ =\ {(\frac{1}{2})}^{3}\)
    With the same base, the exponents must be equal.	\(\ x\ =\ 3\ \text{Therefore,}\ {\text{log}}_{\frac{1}{2}}\frac{1}{8}=3\)

11. Find the value of \(x:\) ⓐ \({\text{log}}_{x}64=2\) ⓑ \({\text{log}}_{5}x=3\) ⓒ \({\text{log}}_{\frac{1}{2}}\frac{1}{4}=x\)

    Sýna svarið

    
    ⓐ \(x=8\) ⓑ \(x=125\) ⓒ \(x=2\)

12. Find the value of \(x:\) ⓐ \({\text{log}}_{x}81=2\) ⓑ \({\text{log}}_{3}x=5\) ⓒ \({\text{log}}_{\frac{1}{3}}\frac{1}{27}=x\)

    Sýna svarið

    
    ⓐ
    \(x=9\) ⓑ \(x=243\) ⓒ \(x=3\)

13. Find the exact value of each logarithm without using a calculator: ⓐ \({\text{log}}_{5}25,\) ⓑ \({\text{log}}_{9}3,\) and ⓒ \({\text{log}}_{2}\frac{1}{16}.\)

    Sýna svarið

    ⓐ
    

    	\({\text{log}}_{5}25\)
    5 to what power will be 25?	\({\text{log}}_{5}25\ =\ 2\)
    Or
    Set the expression equal to \(x\).	\({\text{log}}_{5}25\ =\ x\)
    Change to exponential form.	\(\ {5}^{x}\ =\ 25\)
    Rewrite 25 as \({5}^{2}\).	\(\ {5}^{x}\ =\ {5}^{2}\)
    With the same base the exponents must be equal.	\(\ x\ =\ 2\ \text{Therefore,}\ {\text{log}}_{5}25=2.\)

    ⓑ
    

    	\({\text{log}}_{9}3\)
    Set the expression equal to \(x\).	\({\text{log}}_{9}3\ =\ x\)
    Change to exponential form.	\(\ {9}^{x}\ =\ 3\)
    Rewrite 9 as \({3}^{2}\).	\({({3}^{2})}^{x}\ =\ {3}^{1}\)
    Simplify the exponents.	\(\ {3}^{2x}\ =\ {3}^{1}\)
    With the same base the exponents must be equal.	\(\ 2x\ =\ 1\)
    Solve the equation.	\(\ x\ =\ \frac{1}{2}\ \text{Therefore,}\ {\text{log}}_{9}3=\frac{1}{2}.\)

    ⓒ
    

    	\({\text{log}}_{2}\frac{1}{16}\)
    Set the expression equal to \(x\).	\({\text{log}}_{2}\frac{1}{16}\ =\ x\)
    Change to exponential form.	\(\ {2}^{x}\ =\ \frac{1}{16}\)
    Rewrite 16 as \({2}^{4}\).	\(\ {2}^{x}\ =\ \frac{1}{{2}^{4}}\)
    	\(\ {2}^{x}\ =\ {2}^{-4}\)
    With the same base the exponents must be equal.	\(\ x\ =\ -4\ \text{Therefore,}\ {\text{log}}_{2}\frac{1}{16}=-4.\)

14. Find the exact value of each logarithm without using a calculator: ⓐ \({\text{log}}_{12}144\) ⓑ \({\text{log}}_{4}2\) ⓒ \({\text{log}}_{2}\frac{1}{32}\)

    Sýna svarið

    ⓐ
    2 ⓑ \(\frac{1}{2}\) ⓒ \(-5\)

15. Find the exact value of each logarithm without using a calculator: ⓐ \({\text{log}}_{9}81\) ⓑ \({\text{log}}_{8}2\) ⓒ \({\text{log}}_{3}\frac{1}{9}\)

    Sýna svarið

    ⓐ 2 ⓑ \(\frac{1}{3}\) ⓒ \(-2\)

16. Graph \(y={\text{log}}_{2}x.\)

    Sýna svarið

    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.

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

17. Graph: \(y={\text{log}}_{3}x.\)

    Sýna svarið

    
    

18. Graph: \(y={\text{log}}_{5}x.\)

    Sýna svarið

    
    

19. Graph \(y={\text{log}}_{\frac{1}{3}}x.\)

    Sýna svarið

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

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

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

20. Graph: \(y={\text{log}}_{\frac{1}{2}}x.\)

    Sýna svarið

    
    

21. Graph: \(y={\text{log}}_{\frac{1}{4}}x.\)

    Sýna svarið

    
    

22. Solve: ⓐ \({\text{log}}_{a}49=2\) and ⓑ \(\text{ln}\ x=3.\)

    Sýna svarið

    ⓐ
    

    	\({\log }_{a}49\ =\ 2\)
    Rewrite in exponential form.	\(\ {a}^{2}\ =\ 49\)
    Solve the equation using the square root property.	\(\ a\ =\ \pm 7\)
    The base cannot be negative, so we eliminate \(a=-7.\)	\(a=7,\ a=-7\)
    Check.
    \(\begin{array}{llll}a=7 & {\log }_{a}49 & = & 2 \\ & {\log }_{7}49 & \overset{?}{=} & 2 \\ & {7}^{2} & \overset{?}{=} & 49 \\ & 49 & = & 49✓\end{array}\)

    ⓑ
    

    	\(\ \text{ln}\ x\ =\ 3\)
    Rewrite in exponential form.	\(\ {e}^{3}\ =\ x\)
    Check.
    \(\begin{array}{llll} \\ x={e}^{3} & \text{ln}\ x & = & 3 \\ & \text{ln}\ {e}^{3} & \overset{?}{=} & 3 \\ & {e}^{3} & = & {e}^{3}✓\end{array}\)

23. Solve: ⓐ \({\text{log}}_{a}121=2\) ⓑ \(\text{ln}\ x=7\)

    Sýna svarið

    
    ⓐ
    \(a=11\)
    ⓑ \(x={e}^{7}\)

24. Solve: ⓐ \({\text{log}}_{a}64=3\) ⓑ \(\text{ln}\ x=9\)

    Sýna svarið

    
    ⓐ
    \(a=4\)
    ⓑ \(x={e}^{9}\)

25. Solve: ⓐ \({\text{log}}_{2}(3x-5)=4\) and ⓑ \(\text{ln}\ {e}^{2x}=4.\)

    Sýna svarið

    ⓐ
    

    	\({\log }_{2}(3x-5)\ =\ 4\\)
    Rewrite in exponential form.	\({2}^{4}\ =\ 3x-5\)
    Simplify.	\(16\ =\ 3x-5\)
    Solve the equation.	\(21\ =\ 3x\\)
    	\(7\ =\ x\\)
    Check.
    \(\begin{array}{llll}x=7 & {\log }_{2}(3x-5) & = & 4 \\ & {\log }_{2}(3⋅7-5) & \overset{?}{=} & 4 \\ & {\log }_{2}(16) & \overset{?}{=} & 4 \\ & {2}^{4} & \overset{?}{=} & 16 \\ & 16 & = & 16✓\ \end{array}\)

    ⓑ
    

    	\(\ln \ {e}^{2x}\ =\ 4\\)
    Rewrite in exponential form.	\({e}^{4}\ =\ {e}^{2x}\)
    Since the bases are the same the exponents are equal.	\(4\ =\ 2x\\)
    Solve the equation.	\(2\ =\ x\\)
    Check.
    \(\begin{array}{llll}x=2 & \ln \ {e}^{2x} & = & 4 \\ & \ln \ {e}^{2\cdot 2} & \overset{?}{=} & 4 \\ & \ln \ {e}^{4} & \overset{?}{=} & 4 \\ & {e}^{4} & = & {e}^{4}✓\end{array}\)

26. Solve: ⓐ \({\text{log}}_{2}(5x-1)=6\) ⓑ \(\text{ln}\ {e}^{3x}=6\)

    Sýna svarið

    
    ⓐ
    \(x=13\)
    ⓑ \(x=2\)

27. Solve: ⓐ \({\text{log}}_{3}(4x+3)=3\) ⓑ \(\text{ln}\ {e}^{4x}=4\)

    Sýna svarið

    
    ⓐ
    \(x=6\)
    ⓑ \(x=1\)

28. 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 \({10}^{-2}\) watts per square inch?

    Sýna svarið

    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.

29. What is the decibel level of one of the new quiet dishwashers with intensity \({10}^{-7}\) watts per square inch?

    Sýna svarið

    The quiet dishwashers have a decibel level of 50 dB.

30. What is the decibel level heavy city traffic with intensity \({10}^{-3}\) watts per square inch?

    Sýna svarið

    The decibel level of heavy traffic is 90 dB.

31. In 1906, San Francisco experienced an intense earthquake with a magnitude of 7.8 on the Richter scale. Over 80% of the city was destroyed by the resulting fires. In 2014, Los Angeles experienced a moderate earthquake that measured 5.1 on the Richter scale and caused $108 million dollars of damage. Compare the intensities of the two earthquakes.

    Sýna svarið

    To compare the intensities, we first need to convert the magnitudes to intensities using the log formula. Then we will set up a ratio to compare the intensities.

    Convert the magnitudes to intensities.	\(\ R=\text{log}\ I\)
    \(\ \text{1906 earthquake}\)	\(7.8=\text{log}\ I\)
    \(\ \text{Convert to exponential form.}\)	\(\ I={10}^{7.8}\)
    \(\ \text{2014 earthquake}\)	\(5.1=\text{log}\ I\)
    \(\ \text{Convert to exponential form.}\)	\(\ I={10}^{5.1}\)
    Form a ratio of the intensities.	\(\frac{\text{Intensity}\ \text{for}\ 1906}{\text{Intensity}\ \text{for}\ 2014}\)
    Substitute in the values.	\(\ \frac{{10}^{7.8}}{{10}^{5.1}}\)
    Divide by subtracting the exponents.	\(\ {10}^{2.7}\)
    Evaluate.	\(\ 501\)
    	The intensity of the 1906 earthquake was about 501 times the intensity of the 2014 earthquake.

32. In 1906, San Francisco experienced an intense earthquake with a magnitude of 7.8 on the Richter scale. In 1989, the Loma Prieta earthquake also affected the San Francisco area, and measured 6.9 on the Richter scale. Compare the intensities of the two earthquakes.

    Sýna svarið

    The intensity of the 1906 earthquake was about 8 times the intensity of the 1989 earthquake.

33. In 2014, Chile experienced an intense earthquake with a magnitude of 8.2 on the Richter scale. In 2014, Los Angeles also experienced an earthquake which measured 5.1 on the Richter scale. Compare the intensities of the two earthquakes.

    Sýna svarið

    The intensity of the earthquake in Chile was about 1,259 times the intensity of the earthquake in Los Angeles.

34. \({4}^{2}=16\)

35. \({2}^{5}=32\)

    Sýna svarið

    \({\text{log}}_{2}32=5\)

36. \({3}^{3}=27\)

37. \({5}^{3}=125\)

    Sýna svarið

    \({\text{log}}_{5}125=3\)

38. \({10}^{3}=1000\)

39. \({10}^{-2}=\frac{1}{100}\)

    Sýna svarið

    \(\text{log}\frac{1}{100}=-2\)

40. \({x}^{\frac{1}{2}}=\sqrt{3}\)

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

i

imaginary unit
i² = −1.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Evaluate and Graph Logarithmic Functions

1. Convert between exponential and logarithmic form
2. Evaluate logarithmic functions
3. Graph Logarithmic functions
4. Solve logarithmic equations
5. Use logarithmic models in applications

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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Leysa

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

Parts of this page are adapted from OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value