Simplifying Log Expressions 3
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Question 1 of 5
1. Question
Simplify and solve`log_8 4-log_8 32+log_8 64`- (1)
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x}-\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$Compress the expression into one logarithm$$\log_\color{#9a00c7}{8} \color{#00880A}{4}-\log_8 32+\log_\color{#9a00c7}{8} \color{#e65021}{64}$$ `=` $$\log_\color{#9a00c7}{8} \color{#00880A}{(4)}\color{#e65021}{(64)}-\log_2 18$$ `log_b xy=log_b x+log_b y` `=` $$\log_8 256-\log_8 32$$ `=` $$\log_\color{#9a00c7}{8} \color{#00880A}{256}-\log_\color{#9a00c7}{8} \color{#e65021}{32}$$ `=` $$\log_\color{#9a00c7}{8} \frac{\color{#00880A}{256}}{\color{#e65021}{32}}$$ $$log_b \frac{x}{y}=log_b x-\log_b y$$ `=` `log_8 8` `=` `1` `log_b b=1` `1` -
Question 2 of 5
2. Question
Simplify and solve`log_10 250+log_10 16-log_10 4`- (3)
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x}-\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$Compress the expression into one logarithm$$\log_\color{#9a00c7}{10} \color{#00880A}{250}+\log_\color{#9a00c7}{10} \color{#e65021}{16}-\log_{10} 4$$ `=` $$\log_\color{#9a00c7}{10} \color{#00880A}{(250)}\color{#e65021}{(16)}-\log_{10} 4$$ `log_b xy=log_b x+log_b y` `=` $$\log_{10} 4000-\log_{10} 4$$ `=` $$\log_\color{#9a00c7}{10} \color{#00880A}{4000}-\log_\color{#9a00c7}{10} \color{#e65021}{4}$$ `=` $$\log_\color{#9a00c7}{10} \frac{\color{#00880A}{4000}}{\color{#e65021}{4}}$$ $$log_b \frac{x}{y}=log_b x-\log_b y$$ `=` `log_(10) 1000` Simplify the logarithm`log_(10) 1000` `=` $$\log_{10} \color{#CC0000}{10^3}$$ $$1000=10^3$$ `=` $$\color{#004ec4}{3}\log_{10} 10$$ `log_b x^p=p log_b x` `=` $$3(\color{#9a00c7}{1})$$ `log_b b=1` `=` `3` `3` -
Question 3 of 5
3. Question
Simplify and solve`3log_2 4+1/2log_2 81-log_2 18`- (5)
Hint
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x}-\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$Remove the coefficients from the logarithms so they can be combinedFirst term:$$3\log_2 4+\frac{1}{2}\log_2 81-\log_2 18$$ `=` $$\log_2 4^\color{#004ec4}{3}+\frac{1}{2}\log_2 81-\log_2 18$$ `log_b x^p=p log_b x` `=` $$\log_2 64+\frac{1}{2}\log_2 81-\log_2 18$$ Second term:`=` $$\log_2 64+\frac{1}{2}\log_2 81-\log_2 18$$ `=` $$\log_2 64+\log_2 81^\color{#004ec4}{\frac{1}{2}}-\log_2 18$$ `log_b x^p=p log_b x` `=` $$\log_2 64+\log_2 \color{#CC0000}{\sqrt{81}}-\log_2 18$$ Change the exponent into a surd `=` `log_2 64+log_2 9-log_2 18` Compress the expression into one logarithm$$\log_\color{#9a00c7}{2} \color{#00880A}{64}+\log_\color{#9a00c7}{2} \color{#e65021}{9}-\log_2 18$$ `=` $$\log_\color{#9a00c7}{2} \color{#00880A}{(64)}\color{#e65021}{(9)}-\log_2 18$$ `log_b xy=log_b x+log_b y` `=` `log_2 576-log_2 18` `=` $$\log_\color{#9a00c7}{2} \color{#00880A}{576}-\log_\color{#9a00c7}{2} \color{#e65021}{18}$$ `=` $$\log_\color{#9a00c7}{2} \frac{\color{#00880A}{576}}{\color{#e65021}{18}}$$ $$log_b \frac{x}{y}=log_b x-\log_b y$$ `=` `log_2 32` Simplify the logarithm`log_2 32` `=` $$\log_2 \color{#CC0000}{2^5}$$ $$32=2^5$$ `=` $$\color{#004ec4}{5}\log_2 2$$ `log_b x^p=p log_b x` `=` `5(``1``)` `log_b b=1` `=` `5` `5` -
Question 4 of 5
4. Question
Simplify and solve`(log_a x^3-log_a x^2)/log_a sqrtx`- (2)
Hint
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Laws of Logarithms
$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$Compress the expression into one logarithm$$\frac{\log_a x^3-\log_a x^2}{\log_a \sqrt{x}}$$ `=` $$\frac{\log_a x^3-\log_a x^2}{\log_a x^\color{#CC0000}{\frac{1}{2}}}$$ Change the surd into an exponent `=` $$\frac{\color{#004ec4}{3}\log_a x-\color{#004ec4}{2}\log_a x}{\color{#004ec4}{\frac{1}{2}}\log_a x}$$ `log_b x^p=p log_b x` `=` $$\frac{1\log_a x}{\frac{1}{2}\log_a x}$$ `=` $$\frac{1}{\frac{1}{2}}$$ `(log_a x)/(log_a x)=1` `=` `2` `2` -
Question 5 of 5
5. Question
Simplify and solve`(12log_a sqrta-log_a (1/(a^2)))/log_a a^2`- (4)
Hint
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x}-\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$Compress the expression into one logarithm$$\frac{12\log_a \sqrt{a}-\log_a \frac{1}{a^2}}{\log_a a^2}$$ `=` $$\frac{12\log_a \sqrt{a}-\log_a \color{#CC0000}{a^{-2}}}{\log_a a^2}$$ Reciprocate `1/a^2` `=` $$\frac{\color{#004ec4}{12}\log_a a^\color{#CC0000}{\frac{1}{2}}-\log_a a^{-2}}{\log_a a^2}$$ Change the surd into an exponent `=` $$\frac{\log_a a^{(\color{#004ec4}{12})(\frac{1}{2})}-\log_a a^{-2}}{\log_a a^2}$$ `log_b x^p=p log_b x` `=` $$\frac{\log_a a^6-\log_a a^{-2}}{\log_a a^2}$$ `=` $$\frac{\log_\color{#9a00c7}{a} \color{#00880A}{a^6}-\log_\color{#9a00c7}{a} \color{#e65021}{a^{-2}}}{\log_a a^2}$$ `=` $$\frac{\log_\color{#9a00c7}{a} \frac{\color{#00880A}{a^6}}{\color{#e65021}{a^{-2}}}}{\log_a a^2}$$ $$log_b \frac{x}{y}=log_b x-\log_b y$$ `=` $$\frac{\log_a a^8}{\log_a a^2}$$ `=` $$\frac{\color{#004ec4}{8}\log_a a}{\color{#004ec4}{2}\log_a a}$$ `log_b x^p=p log_b x` `=` $$\frac{8}{2}$$ `(log_a x)/(log_a x)=1` `=` `4` `4`
Quizzes
- Converting Between Logarithmic and Exponent Form 1
- Converting Between Logarithmic and Exponent Form 2
- Evaluating Logarithms 1
- Evaluating Logarithms 2
- Evaluating Logarithms 3
- Expanding Log Expressions
- Simplifying Log Expressions 1
- Simplifying Log Expressions 2
- Simplifying Log Expressions 3
- Change Of Base Formula
- Logarithmic Equations 1
- Logarithmic Equations 2
- Logarithmic Equations 3
- Solving Exponential Equations