10.4 Use the Properties of Logarithms

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$\begin{array}{cccccc}& & & {\text{log}}_{8}1\hfill & & \\ \text{Use the property,}{\text{log}}_{a}1=0.\hfill & & & 0\hfill & & {\text{log}}_{8}1=0\hfill \end{array}$

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$\begin{array}{cccccc}& & & {\text{log}}_{6}6\hfill & & \\ \text{Use the property,}{\text{log}}_{a}a=1.\hfill & & & 1\hfill & & {\text{log}}_{6}6=1\hfill \end{array}$

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$\begin{array}{cccc}& & & 4^{{\text{log}}_{4}9}\hfill \\ \text{Use the property,}{a}^{{\text{log}}_{a}x}=x.\hfill & & & 9\hfill & 4^{{\text{log}}_{4}9}=9\hfill \end{array}$

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$\begin{array}{cccc}& & & {\text{log}}_3{3}^5\hfill \\ \text{Use the property,}{a}^{{\text{log}}_{a}x}=x.\hfill & & & 5\hfill & {\text{log}}_3{3}^5=5\hfill \end{array}$

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$\begin{array}{cccc}& & & {\text{log}}_{3}7x\hfill \\ \text{Use the Product Property,}{\text{log}}_a\left(M·N\right)={\text{log}}_{a}M+{\text{log}}_{a}N.\hfill & & & {\text{log}}_{3}7+{\text{log}}_{3}x\hfill \\ \\ \\ & & & {\text{log}}_{3}7x={\text{log}}_{3}7+{\text{log}}_{3}x\hfill \end{array}$

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$\begin{array}{cccc}& & & {\text{log}}_{4}64xy\hfill \\ \text{Use the Product Property,}{\text{log}}_a\left(M·N\right)={\text{log}}_{a}M+{\text{log}}_{a}N.\hfill & & & {\text{log}}_{4}64+{\text{log}}_{4}x+{\text{log}}_{4}y\hfill \\ \\ \\ \\ \text{Simplify by evaluating}{\text{log}}_{4}64.\hfill & & & 3+{\log }_{4}x+{\log }_{4}y\hfill \\ & & & {\text{log}}_{4}64xy=3+{\text{log}}_{4}x+{\text{log}}_{4}y\hfill \end{array}$

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$\begin{array}{cccc}& & & {\text{log}}_5\frac{5}{7}\hfill \\ \text{Use the Quotient Property,}{\text{log}}_a\frac{M}{N}={\text{log}}_{a}M-{\text{log}}_{a}N.\hfill & & & {\text{log}}_{5}5-{\text{log}}_{5}7\hfill \\ \text{Simplify.}\hfill & & & 1-{\text{log}}_{5}7\hfill \\ \\ \\ & & & {\text{log}}_5\frac{5}{7}=1-{\text{log}}_{5}7\hfill \end{array}$

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$\begin{array}{cccc}& & & \text{log}\frac{x}{100}\hfill \\ \text{Use the Quotient Property,}{\text{log}}_a\frac{M}{N}={\text{log}}_{a}M-{\text{log}}_{a}N.\hfill & & & \text{log}x-\text{log}100\hfill \\ \text{Simplify.}\hfill & & & \text{log}x-2\hfill \\ \\ \\ & & & \text{log}\frac{x}{100}=\text{log}x-2\hfill \end{array}$

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$\begin{array}{cccc}& & & {\text{log}}_5{4}^3\hfill \\ \text{Use the Power Property,}{\text{log}}_a{M}^p=p{\text{log}}_{a}M.\hfill & & & 3{\text{log}}_{5}4\hfill \\ \\ \\ & & & {\text{log}}_5{4}^3=3{\text{log}}_{5}4\hfill \end{array}$

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$\begin{array}{cccc}& & & \text{log}{x}^{10}\hfill \\ \text{Use the Power Property,}{\text{log}}_a{M}^p=p{\text{log}}_{a}M.\hfill & & & 10\text{log}x\hfill \\ \\ \\ & & & \text{log}{x}^{10}=10\text{log}x\hfill \end{array}$

$\begin{array}{cccc}& & & {\text{log}}_4\left(2x^3{y}^2\right)\hfill \\ \text{Use the Product Property,}{\text{log}}_{a}M·N={\text{log}}_{a}M+{\text{log}}_{a}N.\hfill & & & {\text{log}}_{4}2+{\text{log}}_4{x}^3+{\text{log}}_4{y}^2\hfill \\ \text{Use the Power Property,}{\text{log}}_a{M}^p=p{\text{log}}_{a}M,\text{on the last two terms.}\hfill & & & {\text{log}}_{4}2+3{\text{log}}_{4}x+2{\text{log}}_{4}y\hfill \\ \text{Simplify.}\hfill & & & \frac{1}{2}+3{\text{log}}_{4}x+2{\text{log}}_{4}y\hfill \\ \\ \\ & & & {\text{log}}_4\left(2x^3{y}^2\right)=\frac{1}{2}+3{\text{log}}_{4}x+2{\text{log}}_{4}y\hfill \end{array}$

$\begin{array}{cccc}& & & {\text{log}}_2\sqrt[4]{\frac{x^3}{3y^{2}z}}\hfill \\ \text{Rewrite the radical with a rational exponent.}\hfill & & & {\text{log}}_2{\left(\frac{x^3}{3y^{2}z}\right)}^{\frac{1}{4}}\hfill \\ \\ \\ \text{Use the Power Property,}{\text{log}}_a{M}^p=p{\text{log}}_{a}M.\hfill & & & \frac{1}{4}{\text{log}}_2\left(\frac{x^3}{3y^{2}z}\right)\hfill \\ \\ \\ \text{Use the Quotient Property,}{\text{log}}_{a}M·N={\text{log}}_{a}M-{\text{log}}_{a}N.\hfill & & & \frac{1}{4}\left({\text{log}}_2\left(x^3\right)-{\text{log}}_2\left(3y^{2}z\right)\right)\hfill \\ \\ \\ \begin{array}{c}\text{Use the Product Property,}\hfill \\ {\text{log}}_{a}M·N={\text{log}}_{a}M+{\text{log}}_{a}N,\text{in the second term.}\hfill \end{array}\hfill & & & \frac{1}{4}\left({\text{log}}_2\left(x^3\right)-\left({\text{log}}_{2}3+{\text{log}}_2{y}^2+{\text{log}}_{2}z\right)\right)\hfill \\ \\ \\ \begin{array}{c}\text{Use the Power Property,}\hfill \\ {\text{log}}_a{M}^p=p{\text{log}}_{a}M,\text{inside the parentheses.}\hfill \end{array}\hfill & & & \frac{1}{4}\left(3{\text{log}}_{2}x-\left({\text{log}}_{2}3+2{\text{log}}_{2}y+{\text{log}}_{2}z\right)\right)\hfill \\ \\ \\ \text{Simplify by distributing.}\hfill & & & \frac{1}{4}\left(3{\text{log}}_{2}x-{\text{log}}_{2}3-2{\text{log}}_{2}y-{\text{log}}_{2}z\right)\hfill \\ \\ \\ & & & {\text{log}}_2\sqrt[4]{\frac{x^3}{3y^{2}z}}=\frac{1}{4}\left(3{\text{log}}_{2}x-{\text{log}}_{2}3-2{\text{log}}_{2}y-{\text{log}}_{2}z\right)\hfill \end{array}$

$\begin{array}{cccc}\text{The log expressions all have the same base, 4.}\hfill & & & {\text{log}}_{4}3+{\text{log}}_{4}x-{\text{log}}_{4}y\hfill \\ \text{The first two terms are added, so we use the Product Property,}\hfill & & & \\ {\text{log}}_{a}M+{\text{log}}_{a}N={\text{log}}_{a}M·N.\hfill & & & {\text{log}}_{4}3x-{\text{log}}_{4}y\hfill \\ \begin{array}{c}\text{Since the logs are subtracted, we use the Quotient Property,}\hfill \\ {\text{log}}_{a}M-{\text{log}}_{a}N={\text{log}}_a\frac{M}{N}.\hfill \end{array}\hfill & & & {\text{log}}_4\frac{3x}{y}\hfill \\ & & & {\text{log}}_{4}3+{\text{log}}_{4}x-{\text{log}}_{4}y={\text{log}}_4\frac{3x}{y}\hfill \end{array}$

$\begin{array}{cccc}\text{The log expressions have the same base, 3.}\hfill & & & 2{\text{log}}_{3}x+4{\text{log}}_3\left(x+1\right)\hfill \\ \text{Use the Power Property,}{\text{log}}_{a}M+{\text{log}}_{a}N={\text{log}}_{a}M·N.\hfill & & & {\text{log}}_3{x}^2+{\text{log}}_3{\left(x+1\right)}^4\hfill \\ \begin{array}{c}\text{The terms are added, so we use the Product}\hfill \\ \text{Property,}{\text{log}}_{a}M+{\text{log}}_{a}N={\text{log}}_{a}M·N.\hfill \end{array}\hfill & & & {\text{log}}_3{x}^2{\left(x+1\right)}^4\hfill \\ & & & 2{\text{log}}_{3}x+4{\text{log}}_3\left(x+1\right)={\text{log}}_3{x}^2{\left(x+1\right)}^4\hfill \end{array}$