# Section 3.3 – Properties of Logarithms

Section Details.
• Learning the properties of logarithms
• Using the properties of logarithms to evaluate logarithmic expressions
• Using the properties of logarithms to expand a logarithmic expression
• Using the properties of logarithms to condense an expression to the logarithm of a single quantity
• Using the change of base formula

### 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. Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)
1. $$\log_{4} (64x^2)$$

$$3+2\log_4(x)$$

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2. $$\ln \sqrt[3]{ \dfrac{ x^{2}}{x^2-8x-20}}$$

$$\dfrac{1}{3}\left[2\ln(x)-\ln(x-10)-\ln(x+2)\right]$$

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2. Use the properties of logarithms to condense the expression as a single logarithm. (Assume all variables are positive.)
1. $$2 \log_{5} (x-1) + 4 \log_{5} (y)-1$$

$$\log_5\left(\dfrac{(x-1)^2y^4}{5}\right)$$

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2. $$2\ln (6) - \ln (8) - \ln (81)$$

$$\ln\left(\dfrac{1}{18}\right)=-\ln(18)$$

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3. Change $$\displaystyle{\log_{7} (45)}$$ to base 5.

$$\dfrac{\log_5(45)}{\log_5(7)}$$

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4. Change $$\displaystyle{\log_{6} (x)}$$ to base 10.

$$\dfrac{\log(x)}{\log(6)}$$

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