Simplifying Log Expressions 2
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Question 1 of 4
1. Question
Solve`log_5 125-3log_5 25`- (-3)
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Laws of Logarithms
$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$Use the following laws to simplify the logarithmic expression$$\log_b x^\color{#004ec4}{p}$$ `=` $$\color{#004ec4}{p}\log_b x$$ $$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}$$ `=` $$1$$ `log_5 125-3log_5 25` `=` `log_5` `5^3` `-3log_5 25` `125=5^3` `=` `log_5 5^3-3log_5` `5^2` `25=5^2` `=` `3``log_5 5-3(``2``)log_5 5` `log_b x^p=p log_b x` `=` `3(``1``)-6(``1``)` `log_b b=1` `=` `3-6` `=` `-3` `-3` -
Question 2 of 4
2. Question
Solve`3log_10 2+log_10 12.5`- (2)
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$Remove the coefficients from the logarithms so they can be combined$$3\log_{10} 2+\log_{10} 12.5$$ `=` $$\log_{10} 2^\color{#004ec4}{3}+\log_{10} 12.5$$ `log_b x^p=p log_b x` `=` $$\log_{10} 8+\log_{10} 12.5$$ Next, compare the given expression to one of the laws of logarithms and identify corresponding components$$\log_{\color{#9a00c7}{b}} \color{#00880A}{x}$$ `+` $$\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$ $$\log_{\color{#9a00c7}{10}} \color{#00880A}{8}$$ `+` $$\log_{\color{#9a00c7}{10}} \color{#e65021}{12.5}$$ `b` `=` `10` `x` `=` `8` `y` `=` `12.5` Substitute the components into the law of logarithms$$\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$ `=` $$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}$$ $$\log_{\color{#9a00c7}{10}} \color{#00880A}{8} + \log_{\color{#9a00c7}{10}} \color{#e65021}{12.5}$$ `=` $$\log_{\color{#9a00c7}{10}} {\color{#00880A}{(8)}\color{#e65021}{(12.5)}}$$ `=` `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 3 of 4
3. Question
Solve`2log_3 root(3)(3)+log_3 sqrt27`Hint
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Laws of Logarithms
$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$Use the following laws to simplify the logarithmic expression$$\log_b x^\color{#004ec4}{p}$$ `=` $$\color{#004ec4}{p}\log_b x$$ $$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}$$ `=` $$1$$ `2log_3 root(3)(3)+log_3 sqrt27` `=` $$2\log_3 \sqrt[3]{3}+\log_3 \sqrt{\color{#CC0000}{3^3}}$$ `27=3^3` `=` $$2\log_3 3^\color{#CC0000}{\frac{1}{3}}+\log_3 3^\color{#CC0000}{\frac{3}{2}}$$ Change the surds into exponents `=` `1/3``(2)log_3 3+` `3/2``log_3 3` `log_b x^p=p log_b x` `=` `2/3log_3 3+3/2log_3 3` `=` `2/3 xx``1``+3/2 xx``1` `log_b b=1` `=` `2/3+3/2` `=` `4/6+9/6` Find the common denominator `=` `13/6` `=` `2 1/6` `2 1/6` -
Question 4 of 4
4. Question
Solve`log_a ((x^3)/(x+2))+log_a (x+2)`Hint
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$First, compare the given expression to one of the laws of logarithms and identify corresponding components$$\log_{\color{#9a00c7}{b}} \color{#00880A}{x}$$ `+` $$\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$ $$\log_{\color{#9a00c7}{a}} \color{#00880A}{\frac{x^3}{x+2}}$$ `+` $$\log_{\color{#9a00c7}{a}} (\color{#e65021}{x+2})$$ `b` `=` `a` `x` `=` `(x^3)/(x+2)` `y` `=` `x+2` Substitute the components into the law of logarithms$$\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$ `=` $$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}$$ $$\log_{\color{#9a00c7}{a}} \color{#00880A}{\frac{x^3}{x+2}} + \log_{\color{#9a00c7}{a}} (\color{#e65021}{x+2})$$ `=` $$\log_{\color{#9a00c7}{a}} {\color{#00880A}{\frac{x^3}{x+2}}\times{\color{#e65021}{x+2}}}$$ `=` `log_a x^3` `(x+2)/(x+2)=1` Simplify the logarithm`log_a x^3` `=` `3``log_a x` `log_b x^p=p log_b x` `3 log_a x`
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