10.4 Use the Properties of Logarithms

Before you get started, take this readiness quiz.

Evaluate: ⓐ $a^{0}$ ⓑ $a^{1} .$
If you missed this problem, review the reference.

Write with a rational exponent: $\sqrt[3]{x^{2} y} .$
If you missed this problem, review the reference.

Round to three decimal places: 2.5646415.
If you missed this problem, review the reference.

Evaluate using the properties of logarithms: ⓐ $\text{log}_{8} 1$ and ⓑ $\text{log}_{6} 6 .$

Evaluate using the properties of logarithms: ⓐ $\text{log}_{13} 1$ ⓑ $\text{log}_{9} 9 .$

Evaluate using the properties of logarithms: ⓐ $\text{log}_{5} 1$ ⓑ $\text{log}_{7} 7 .$

Evaluate using the properties of logarithms: ⓐ $4^{\text{log}_{4} 9}$ and ⓑ $\text{log}_{3} 3^{5} .$

Evaluate using the properties of logarithms: ⓐ $5^{\text{log}_{5} 15}$ ⓑ $\text{log}_{7} 7^{4} .$

Evaluate using the properties of logarithms: ⓐ $2^{\text{log}_{2} 8}$ ⓑ $\text{log}_{2} 2^{15} .$

Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible: ⓐ $\text{log}_{3} 7 x$ and ⓑ $\text{log}_{4} 64 x y .$

Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

ⓐ $\text{log}_{3} 3 x$ ⓑ $\text{log}_{2} 8 x y$

Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

ⓐ $\text{log}_{9} 9 x$ ⓑ $\text{log}_{3} 27 x y$

Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.
ⓐ $\text{log}_{5} \frac{5}{7}$ and ⓑ $\text{log} \frac{x}{100}$

Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.

ⓐ $\text{log}_{4} \frac{3}{4}$ ⓑ $\text{log} \frac{x}{1000}$

Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.

ⓐ $\text{log}_{2} \frac{5}{4}$ ⓑ $\text{log} \frac{10}{y}$

Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.
ⓐ $\text{log}_{5} 4^{3}$ and ⓑ $\text{log} x^{10}$

Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.

ⓐ $\text{log}_{7} 5^{4}$ ⓑ $\text{log} x^{100}$

Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.

ⓐ $\text{log}_{2} 3^{7}$ ⓑ $\text{log} x^{20}$

Use the Properties of Logarithms to expand the logarithm $\text{log}_{4} \left(2 x^{3} y^{2}\right)$. Simplify, if possible.

Use the Properties of Logarithms to expand the logarithm $\text{log}_{2} \left(5 x^{4} y^{2}\right)$. Simplify, if possible.

Use the Properties of Logarithms to expand the logarithm $\text{log}_{3} \left(7 x^{5} y^{3}\right)$. Simplify, if possible.

Use the Properties of Logarithms to expand the logarithm $\text{log}_{2} \sqrt[4]{\frac{x^{3}}{3 y^{2} z}}$. Simplify, if possible.

Use the Properties of Logarithms to expand the logarithm $\text{log}_{4} \sqrt[5]{\frac{x^{4}}{2 y^{3} z^{2}}}$. Simplify, if possible.

Use the Properties of Logarithms to expand the logarithm $\text{log}_{3} \sqrt[3]{\frac{x^{2}}{5 y^{} z}}$. Simplify, if possible.

Use the Properties of Logarithms to condense the logarithm $\text{log}_{4} 3 + \text{log}_{4} x - \text{log}_{4} y$. Simplify, if possible.

Use the Properties of Logarithms to condense the logarithm $\text{log}_{2} 5 + \text{log}_{2} x - \text{log}_{2} y$. Simplify, if possible.

Use the Properties of Logarithms to condense the logarithm $\text{log}_{3} 6 - \text{log}_{3} x - \text{log}_{3} y$. Simplify, if possible.

Use the Properties of Logarithms to condense the logarithm $2 \text{log}_{3} x + 4 \text{log}_{3} \left(x + 1\right)$. Simplify, if possible.

Use the Properties of Logarithms to condense the logarithm $3 \text{log}_{2} x + 2 \text{log}_{2} \left(x - 1\right)$. Simplify, if possible.

Use the Properties of Logarithms to condense the logarithm $2 \text{log} x + 2 \text{log} \left(x + 1\right)$. Simplify, if possible.

Rounding to three decimal places, approximate $\text{log}_{3} 42 .$

Rounding to three decimal places, approximate $\text{log}_{5} 46 .$

Use the properties of logarithms to evaluate.

ⓐ $\text{log}_{12} 1$ ⓑ $\text{ln} e$

Use the properties of logarithms to evaluate.

ⓐ $5^{\text{log}_{5} 10}$ ⓑ $\text{log}_{4} 4^{10}$

Use the properties of logarithms to evaluate.

ⓐ $6^{\text{log}_{6} 15}$ ⓑ $\text{log}_{8} 8^{−4}$

Use the properties of logarithms to evaluate.

ⓐ $10^{\text{log} \sqrt{3}}$ ⓑ $\text{log} 10^{−1}$

Use the properties of logarithms to evaluate.

ⓐ $e^{\text{ln} 3}$ ⓑ $\text{ln} e^{7}$

Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$\text{log}_{5} 8 y$

Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$\text{log}_{3} 81 x y$

Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$\text{log} 1000 y$

Use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$\text{log}_{6} \frac{5}{6}$

Use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$\text{log}_{5} \frac{125}{x}$

Use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$\text{log} \frac{10,000}{y}$

Use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$\text{ln} \frac{e^{4}}{16}$

Use the Power Property of Logarithms to expand each. Simplify if possible.

$\text{log}_{2} x^{5}$

Use the Power Property of Logarithms to expand each. Simplify if possible.

$\text{log} x^{−3}$

Use the Power Property of Logarithms to expand each. Simplify if possible.

$\text{log}_{5} \sqrt[3]{x}$

Use the Power Property of Logarithms to expand each. Simplify if possible.

$\text{ln} x^{\sqrt[3]{4}}$

Use the Properties of Logarithms to expand the logarithm. Simplify if possible.

$\text{log}_{2} \left(3 x^{5} y^{3}\right)$

Use the Properties of Logarithms to expand the logarithm. Simplify if possible.

$\text{log}_{5} \left(\sqrt[4]{21} y^{3}\right)$

Use the Properties of Logarithms to expand the logarithm. Simplify if possible.

$\text{log}_{5} \frac{4 a b^{3} c^{4}}{d^{2}}$

Use the Properties of Logarithms to expand the logarithm. Simplify if possible.

$\text{log}_{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}}$

Use the Properties of Logarithms to expand the logarithm. Simplify if possible.

$\text{log}_{3} \frac{\sqrt{3 x + 2 y^{2}}}{5 z^{2}}$

Use the Properties of Logarithms to expand the logarithm. Simplify if possible.

$\text{log}_{5} \sqrt[3]{\frac{3 x^{2}}{4 y^{3} z}}$

Use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$\text{log} 4 + \text{log} 25$

Use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$\text{log}_{3} 36 - \text{log}_{3} 4$

Use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$\text{log}_{2} 5 - \text{log}_{2} \left(x - 1\right)$

Use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$\text{log}_{5} 2 - \text{log}_{5} x - \text{log}_{5} y$

Use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$6 \text{log}_{3} x + 9 \text{log}_{3} y$

Use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$\text{log} \left(x^{2} + 2 x + 1\right) - 2 \text{log} \left(x + 1\right)$

Use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$3 \text{ln} x + 4 \text{ln} y - 2 \text{ln} z$

Use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$2 \text{log} \left(2 x + 3\right) + \frac{1}{2} \text{log} \left(x + 1\right)$

Use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.

$\text{log}_{5} 46$

Use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.

$\text{log}_{15} 93$

Use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.

$\text{log}_{\sqrt{3}} 21$

Write the Power Property in your own words. Does it apply to each of the following? $\text{log}_{a} x^{p} ,$$\left(\text{log}_{a} x\right)^{r} .$ Why or why not?

Explain how to find the value of $\text{log}_{7} 15$ using your calculator.