Simplifying Log Expressions 1
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Question 1 of 4
1. Question
Solve`log_6 12+log_6 3`- (2)
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$First, compare the given expression to one of the laws of logarithms and identify corresponding components$$\log_{\color{#9a00c7}{b}} \color{#00880A}{x}$$ `+` $$\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$ $$\log_{\color{#9a00c7}{6}} \color{#00880A}{12}$$ `+` $$\log_{\color{#9a00c7}{6}} \color{#e65021}{3}$$ `b` `=` `6` `x` `=` `12` `y` `=` `3` Substitute the components into the law of logarithms$$\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$ `=` $$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}$$ $$\log_{\color{#9a00c7}{6}} \color{#00880A}{12} + \log_{\color{#9a00c7}{6}} \color{#e65021}{3}$$ `=` $$\log_{\color{#9a00c7}{6}} {\color{#00880A}{(12)}\color{#e65021}{(3)}}$$ `=` `log_6 36` Let the simplified logarithm be equal to `a``a` `=` `log_6 36` `6^a` `=` `36` Convert to exponent form `6^a` `=` `6^2` `36=6^2` `a` `=` `2` Equate the exponents since the bases are equal `2` -
Question 2 of 4
2. Question
Solve`log_2 48-log_2 3`- (4)
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x}-\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$First, compare the given expression to one of the laws of logarithms and identify corresponding components$$\log_{\color{#9a00c7}{b}} \color{#00880A}{x}$$ `-` $$\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$ $$\log_{\color{#9a00c7}{2}} \color{#00880A}{48}$$ `-` $$\log_{\color{#9a00c7}{2}} \color{#e65021}{3}$$ `b` `=` `2` `x` `=` `48` `y` `=` `3` Substitute the components into the law of logarithms$$\log_{\color{#9a00c7}{b}} \color{#00880A}{x}-\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$ `=` $$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}$$ $$\log_{\color{#9a00c7}{2}} \color{#00880A}{48}-\log_{\color{#9a00c7}{2}} \color{#e65021}{3}$$ `=` $$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{48}}{\color{#e65021}{3}}$$ `=` `log_2 16` Let the simplified logarithm be equal to `a``a` `=` `log_2 16` `2^a` `=` `16` Convert to exponent form `2^a` `=` `2^4` `16=2^4` `a` `=` `4` Equate the exponents since the bases are equal `4` -
Question 3 of 4
3. Question
Solve`log_10 25+log_10 4`- (2)
Hint
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Incorrect
Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$First, compare the given expression to one of the laws of logarithms and identify corresponding components$$\log_{\color{#9a00c7}{b}} \color{#00880A}{x}$$ `+` $$\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$ $$\log_{\color{#9a00c7}{10}} \color{#00880A}{25}$$ `+` $$\log_{\color{#9a00c7}{10}} \color{#e65021}{4}$$ `b` `=` `10` `x` `=` `25` `y` `=` `4` Substitute the components into the law of logarithms$$\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$ `=` $$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}$$ $$\log_{\color{#9a00c7}{10}} \color{#00880A}{25} + \log_{\color{#9a00c7}{10}} \color{#e65021}{4}$$ `=` $$\log_{\color{#9a00c7}{10}} {\color{#00880A}{(25)}\color{#e65021}{(4)}}$$ `=` `log_10 100` Simplify the logarithm`log_10 100` `=` `log_10``10^2` `100=10^2` `=` $$\color{#004ec4}{2}\log_{10} 10$$ `log_b x^p=p log_b x` `=` `2(``1``)` `log_b b=1` `=` `2` `2` -
Question 4 of 4
4. Question
Solve`log_5 1000-log_5 8`- (3)
Hint
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Fantastic!
Incorrect
Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x}-\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$First, compare the given expression to one of the laws of logarithms and identify corresponding components$$\log_{\color{#9a00c7}{b}} \color{#00880A}{x}$$ `-` $$\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$ $$\log_{\color{#9a00c7}{5}} \color{#00880A}{1000}$$ `-` $$\log_{\color{#9a00c7}{5}} \color{#e65021}{8}$$ `b` `=` `5` `x` `=` `1000` `y` `=` `8` Substitute the components into the law of logarithms$$\log_{\color{#9a00c7}{b}} \color{#00880A}{x}-\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$ `=` $$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}$$ $$\log_{\color{#9a00c7}{5}} \color{#00880A}{1000}-\log_{\color{#9a00c7}{5}} \color{#e65021}{8}$$ `=` $$\log_{\color{#9a00c7}{5}} \frac{\color{#00880A}{1000}}{\color{#e65021}{8}}$$ `=` `log_5 125` Simplify the logarithm`log_5 125` `=` `log_5` `5^3` `125=5^3` `=` `3``log_5 5` `log_b x^p=p log_b x` `=` `3(``1``)` `log_b b=1` `=` `3` `3`
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