6.5 Logarithmic Properties

Simplify expressions using the properties for exponents.

1. ⓐ Simplify $3 \cdot 2^{x} \cdot 2^{3 x}$
2. ⓑ Simplify $\frac{b^{2} b^{6} b}{b^{4} b^{7}}$
3. ⓒ Simplify $\left(a b^{2}\right)^{3} a^{5} b^{- 6}$

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

Expand $\log_{3} \left(30 x \left(3 x + 4\right)\right) .$

Expand $\log_{b} \left(8 k\right) .$

Expand $\log_{2} \left(\frac{15 x \left(x - 1\right)}{\left(3 x + 4\right) \left(2 - x\right)}\right) .$

Expand $\log_{3} \left(\frac{7 x^{2} + 21 x}{7 x \left(x - 1\right) \left(x - 2\right)}\right) .$

Expand $\log_{2} x^{5} .$

Expand $\ln x^{2} .$

Expand $\log_{3} \left(25\right)$ using the power rule for logs.

Expand $\ln \left(\frac{1}{x^{2}}\right) .$

Rewrite $4 \ln \left(x\right)$ using the power rule for logs to a single logarithm with a leading coefficient of 1.

Rewrite $2 \log_{3} 4$ using the power rule for logs to a single logarithm with a leading coefficient of 1.

Rewrite $\ln \left(\frac{x^{4} y}{7}\right)$ as a sum or difference of logs.

Expand $\log \left(\frac{x^{2} y^{3}}{z^{4}}\right) .$

Expand $\log \left(\sqrt{x}\right) .$

Expand $\ln \left(\sqrt[3]{x^{2}}\right) .$

Expand $\log_{6} \left(\frac{64 x^{3} \left(4 x + 1\right)}{\left(2 x - 1\right)}\right) .$

Expand $\ln \left(\frac{\sqrt{\left(x - 1\right) \left(2 x + 1\right)^{2}}}{\left(x^{2} - 9\right)}\right) .$

Write $\log_{3} \left(5\right) + \log_{3} \left(8\right) - \log_{3} \left(2\right)$ as a single logarithm.

Condense $\log 3 - \log 4 + \log 5 - \log 6.$

Condense $\log_{2} \left(x^{2}\right) + \frac{1}{2} \log_{2} \left(x - 1\right) - 3 \log_{2} \left(\left(x + 3\right)^{2}\right) .$

Rewrite $\log \left(5\right) + 0.5 \log \left(x\right) - \log \left(7 x - 1\right) + 3 \log \left(x - 1\right)$ as a single logarithm.

Rewrite $2 \log x - 4 \log \left(x + 5\right) + \frac{1}{x} \log \left(3 x + 5\right)$ as a single logarithm.

Condense $4 \left(3 \log \left(x\right) + \log \left(x + 5\right) - \log \left(2 x + 3\right)\right) .$

Recall that, in chemistry, $\text{pH} = - \log \left[H^{+}\right] .$ If the concentration of hydrogen ions in a liquid is doubled, what is the effect on pH?

How does the pH change when the concentration of positive hydrogen ions is decreased by half?

Change $\log_{5} 3$ to a quotient of natural logarithms.

Change $\log_{0.5} 8$ to a quotient of natural logarithms.

Evaluate $\log_{2} \left(10\right)$ using the change-of-base formula with a calculator.

Evaluate $\log_{5} \left(100\right)$ using the change-of-base formula.

How does the power rule for logarithms help when solving logarithms with the form $\log_{b} \left(\sqrt[n]{x}\right) ?$

Expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

$\log_{b} \left(7 x \cdot 2 y\right)$

Expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

$\log_{b} \left(\frac{13}{17}\right)$

Expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

$\ln \left(\frac{1}{4^{k}}\right)$

Condense to a single logarithm if possible.

$\ln \left(7\right) + \ln \left(x\right) + \ln \left(y\right)$

Condense to a single logarithm if possible.

$\log_{b} \left(28\right) - \log_{b} \left(7\right)$

Condense to a single logarithm if possible.

$- \log_{b} \left(\frac{1}{7}\right)$

Use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

$\log \left(\frac{x^{15} y^{13}}{z^{19}}\right)$

Use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

$\log \left(\sqrt{x^{3} y^{- 4}}\right)$

Use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

$\log \left(x^{2} y^{3} \sqrt[3]{x^{2} y^{5}}\right)$

Condense each expression to a single logarithm using the properties of logarithms.

$\ln \left(6 x^{9}\right) - \ln \left(3 x^{2}\right)$

Condense each expression to a single logarithm using the properties of logarithms.

$\log \left(x\right) - \frac{1}{2} \log \left(y\right) + 3 \log \left(z\right)$

Rewrite each expression as an equivalent ratio of logs using the indicated base.

$\log_{7} \left(15\right)$ to base $e$

Suppose $\log_{5} \left(6\right) = a$ and $\log_{5} \left(11\right) = b .$ Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of $a$ and $b .$ Show the steps for solving.

$\log_{11} \left(5\right)$

Suppose $\log_{5} \left(6\right) = a$ and $\log_{5} \left(11\right) = b .$ Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of $a$ and $b .$ Show the steps for solving.

$\log_{11} \left(\frac{6}{11}\right)$

Use properties of logarithms to evaluate without using a calculator.

$6 \log_{8} \left(2\right) + \frac{\log_{8} \left(64\right)}{3 \log_{8} \left(4\right)}$

Use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

$\log_{3} \left(22\right)$

Use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

$\log_{6} \left(5.38\right)$

Use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

$\log_{\frac{1}{2}} \left(4.7\right)$

Use the quotient rule for logarithms to find all $x$ values such that $\log_{6} \left(x + 2\right) - \log_{6} \left(x - 3\right) = 1.$ Show the steps for solving.

Prove that $\log_{b} \left(n\right) = \frac{1}{\log_{n} \left(b\right)}$ for any positive integers $b > 1$ and $n > 1.$