10.3 Evaluate and Graph Logarithmic Functions

Before you get started, take this readiness quiz.

Solve: $x^{2} = 81 .$
If you missed this problem, review the reference.

Evaluate: $3^{−2} .$
If you missed this problem, review the reference.

Solve: $2^{4} = 3 x - 5 .$
If you missed this problem, review the reference.

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

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

Convert to exponential form: ⓐ $3 = \text{log}_{4} 64$ ⓑ $0 = \text{log}_{x} 1$ ⓒ $−2 = \text{log}_{10} \frac{1}{100}$

Convert to exponential form: ⓐ $3 = \text{log}_{3} 27$ ⓑ $0 = \text{log}_{3} 1$ ⓒ $−1 = \text{log}_{10} \frac{1}{10}$

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

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

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

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

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

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

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

Solve: ⓐ $\text{log}_{a} 121 = 2$ ⓑ $\text{ln} x = 7$

Solve: ⓐ $\text{log}_{a} 64 = 3$ ⓑ $\text{ln} x = 9$

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

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

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

What is the decibel level of one of the new quiet dishwashers with intensity $10^{−7}$ watts per square inch?

What is the decibel level heavy city traffic with intensity $10^{−3}$ watts per square inch?

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.

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.

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.

Convert from exponential to logarithmic form.

$2^{5} = 32$

Convert from exponential to logarithmic form.

$5^{3} = 125$

Convert from exponential to logarithmic form.

$10^{−2} = \frac{1}{100}$

Convert from exponential to logarithmic form.

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

Convert from exponential to logarithmic form.

$17^{x} = \sqrt[5]{17}$

Convert from exponential to logarithmic form.

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

Convert from exponential to logarithmic form.

$4^{−3} = \frac{1}{64}$

Convert from exponential to logarithmic form.

$e^{3} = x$

Convert each logarithmic equation to exponential form.

$6 = \text{log}_{2} 64$

Convert each logarithmic equation to exponential form.

$5 = \text{log}_{x} 32$

Convert each logarithmic equation to exponential form.

$0 = \text{log}_{7} 1$

Convert each logarithmic equation to exponential form.

$1 = \text{log}_{9} 9$

Convert each logarithmic equation to exponential form.

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

Convert each logarithmic equation to exponential form.

$x = \text{log}_{e} 43$

Find the value of $x$ in each logarithmic equation.

$\text{log}_{x} 121 = 2$

Find the value of $x$ in each logarithmic equation.

$\text{log}_{x} 64 = 3$

Find the value of $x$ in each logarithmic equation.

$\text{log}_{5} x = 3$

Find the value of $x$ in each logarithmic equation.

$\text{log}_{3} x = −5$

Find the value of $x$ in each logarithmic equation.

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

Find the value of $x$ in each logarithmic equation.

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

Find the exact value of each logarithm without using a calculator.

$\text{log}_{6} 36$

Find the exact value of each logarithm without using a calculator.

$\text{log}_{5} 1$

Find the exact value of each logarithm without using a calculator.

$\text{log}_{27} 3$

Find the exact value of each logarithm without using a calculator.

$\text{log}_{\frac{1}{2}} 4$

Find the exact value of each logarithm without using a calculator.

$\text{log}_{3} \frac{1}{27}$

Find the exact value of each logarithm without using a calculator.

$\text{log}_{9} \frac{1}{81}$

Solve each logarithmic equation.

$\text{log}_{a} 81 = 2$

Solve each logarithmic equation.

$\text{log}_{a} 27 = 3$

Solve each logarithmic equation.

$\text{log}_{a} 24 = 3$

Solve each logarithmic equation.

$\text{ln} x = 4$

Solve each logarithmic equation.

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

Solve each logarithmic equation.

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

Solve each logarithmic equation.

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

Solve each logarithmic equation.

$\text{ln} e^{2 x} = 6$

Solve each logarithmic equation.

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

Solve each logarithmic equation.

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

Use a logarithmic model to solve.

What is the decibel level of a whisper with intensity $10^{−10}$ watts per square inch?

Use a logarithmic model to solve.

What is the decibel level of the sound of a garbage disposal with intensity $10^{−2}$ watts per square inch?

Use a logarithmic model to solve.

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.

Explain the difference between common logarithms and natural logarithms.

Explain how to find the $\text{log}_{7} 32$ on your calculator.