 # Section 3.2 – Logarithmic Functions and Their Graphs

Section Details.
• Definition, properties, and terminology for logarithms
• Evaluating logarithms
• The natural log function
• The relationship between logarithms and exponentials
• Using the properties of logarithms to simplify a logarithmic expression
• Graphing a logarithmic function and determining its properties

### Practice Problems

Directions. The following are review problems for the section. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.

1. Simplify each of the following without a calculator:
1. $$\log_{4} (64)$$
2. $$7^{\log_{7} (4)}+2$$
3. $$\displaystyle{\log (10^{-5})}$$
4. $$\log_{11}(3x+5) =\log_{11} (9)$$

a. $$\displaystyle{\log_{4} (64)}=3$$
b. $$\displaystyle{7^{\log_{7} (4)}}+2=6$$
c. $$\displaystyle{\log (10^{-5})}=-5$$
d. $$\displaystyle{\log_{11}(3x+5) =\log_{11} (9)} \implies x=\dfrac{4}{3}$$

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2. Consider the function $$g(x) = -\log_{3}(x+5)+2$$.
1. Describe the transformations of $$f(x) = \log_{3}(x)$$ that yield $$g(x) = -\log_{3}(x+5)+2$$. Then state the domain, $$x$$-intercept, $$y$$-intercept,  and vertical asymptote of the logarithmic function $$f(x)$$.

Transformations: Left 5, reflect over $$x$$-axis, up 2
Domain: $$(-5,\infty)$$
$$x$$-intercept: $$(4,0)$$
$$y$$-intercept: $$(0,-\log_3(5)+2)$$
Vertical asymptote: $$x=-5$$

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2. Choose the graph that matches the function $$g(x) = -\log_{3}(x+5)+2$$. The dark blue graph is the correct graph.

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3. Consider the function $$g(x) =\ln (2x+3)-4$$.
1. Describe the transformations of $$f(x) = \ln (x)$$ that yield $$g(x) =\ln (2x+3)-4$$. Then state the domain, $$x$$-intercept, $$y$$-intercept, and vertical asymptote of the logarithmic function $$f(x)$$.

Transformations: Left 3/2, horizontal shrink by 1/2, down 4
Domain: $$\left(-\dfrac{3}{2},\infty\right)$$
$$x$$-intercept: $$\left(\dfrac{e^4-3}{2}, 0\right)$$
$$y$$-intercept: $$(0,\ln(3)-4)$$
Vertical asymptote: $$x=-\dfrac{3}{2}$$

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2. Choose the graph that matches the function $$g(x) =\ln (2x+3)-4$$. The dark blue graph is the correct graph.

To see the full video page and find related videos, click the following link.