Evaluating Logarithms 2
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Question 1 of 5
1. Question
Solve for `x``log_9 27=x`- `x=` (3/2, 1.5)
Hint
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Exponent Form
$$\color{#00880a}{N}={\color{#9a00c7}{a}}^x$$Logarithmic Form
$$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}=x$$Convert the equation to exponent form by first identifying the components$$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$$ `=` $$x$$ $$\log_{\color{#9a00c7}{9}} \color{#00880a}{27}$$ `=` $$x$$ `N` `=` `27` `a` `=` `9` `x` `=` `x` Substitute the components into the exponent form$$\color{#00880a}{N}$$ `=` $${\color{#9a00c7}{a}}^x$$ $$\color{#00880a}{27}$$ `=` $${\color{#9a00c7}{9}}^x$$ Make sure that only `x` is on the left side`27` `=` `9^x` `27` `=` `(3^2)^x` `9=3^2` `3^3` `=` `3^(2x)` `27=3^3` `3` `=` `2x` Equate the exponents since the bases are equal `3``divide 2` `=` `2x``divide 2` Divide both sides by `2` `3/2` `=` `x` `x` `=` `3/2` `x=3/2` -
Question 2 of 5
2. Question
Solve for `x``log_8 (1/16)=x`- `x=` (-4/3)
Hint
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Exponent Form
$$\color{#00880a}{N}={\color{#9a00c7}{a}}^x$$Logarithmic Form
$$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}=x$$Convert the equation to exponent form by first identifying the components$$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$$ `=` $$x$$ $$\log_{\color{#9a00c7}{8}} \color{#00880a}{\frac{1}{16}}$$ `=` $$x$$ `N` `=` `1/16` `a` `=` `8` `x` `=` `x` Substitute the components into the exponent form$$\color{#00880a}{N}$$ `=` $${\color{#9a00c7}{a}}^x$$ $$\color{#00880a}{\frac{1}{16}}$$ `=` $${\color{#9a00c7}{8}}^x$$ Make sure that only `x` is on the left side`1/16` `=` `8^x` `1/16` `=` `(2^3)^x` `8=2^3` `1/(2^4)` `=` `2^(3x)` `16=2^4` `2^(-4)` `=` `2^(3x)` Reciprocate `1/(2^4)` `-4` `=` `3x` Equate the exponents since the bases are equal `-4``divide3` `=` `3x``divide3` Divide both sides by `3` `-4/3` `=` `x` `x` `=` `-4/3` `x=-4/3` -
Question 3 of 5
3. Question
Solve for `x``log_128 64=x`- `x=` (6/7)
Hint
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Exponent Form
$$\color{#00880a}{N}={\color{#9a00c7}{a}}^x$$Logarithmic Form
$$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}=x$$Convert the equation to exponent form by first identifying the components$$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$$ `=` $$x$$ $$\log_{\color{#9a00c7}{128}} \color{#00880a}{64}$$ `=` $$x$$ `N` `=` `64` `a` `=` `128` `x` `=` `x` Substitute the components into the exponent form$$\color{#00880a}{N}$$ `=` $${\color{#9a00c7}{a}}^x$$ $$\color{#00880a}{64}$$ `=` $${\color{#9a00c7}{128}}^x$$ Make sure that only `x` is on the left side`64` `=` `128^x` `64` `=` `(2^7)^x` `128=2^7` `2^6` `=` `2^(7x)` `64=2^6` `6` `=` `7x` Equate the exponents since the bases are equal `6``divide 7` `=` `7x``divide 7` Divide both sides by `7` `6/7` `=` `x` `x` `=` `6/7` `x=6/7` -
Question 4 of 5
4. Question
Solve for `log_a 6`, given that:`log_a 2=0.3010``log_a 3=0.4771`Round your answer to 4 decimal places- `log_a 6=` (0.7781, .7781)
Hint
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$Expand the given logarithmic expression$$\log_a 6$$ `=` $$\log_a (3)(2)$$ `6=(3)(2)` `=` $$\log_\color{#9a00c7}{a} \color{#00880A}{(3)}\color{#e65021}{(2)}$$ `=` $$\log_\color{#9a00c7}{a} \color{#00880A}{3}+\log_\color{#9a00c7}{a} \color{#e65021}{2}$$ `log_b xy=log_b x+log_b y` Substitute the given values$$\log_a 2$$ `=` $$0.3010$$ $$\log_a 3$$ `=` $$0.4771$$ `log_a 3+log_a 2` `=` `0.4771+0.3010` `=` `0.7781` `0.7781` -
Question 5 of 5
5. Question
Solve for `log_a (4/9)`, given that:`log_a 2=0.3010``log_a 3=0.4771`Round your answer to 4 decimal places- `log_a (4/9)=` (-0.3522)
Hint
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Excellent!
Incorrect
Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x}-\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$Expand the given logarithmic expression$$\log_a \frac{4}{9}$$ `=` $$\log_a \frac{2^2}{9}$$ `4=2^2` `=` $$\log_a \frac{2^2}{3^3}$$ `3=3^2` `=` $$\log_\color{#9a00c7}{a} {\frac{\color{#00880A}{2^2}}{\color{#e65021}{3^2}}}$$ `=` $$\log_\color{#9a00c7}{a} \color{#00880A}{2^2}-\log_\color{#9a00c7}{a} \color{#e65021}{3^2}$$ $$log_b \frac{x}{y}=log_b x-\log_b y$$ `=` $$\color{#004ec4}{2}\log_{a} 2-\color{#004ec4}{2}\log_{a} 3$$ `log_b x^p=p log_b x` Substitute the given values$$\log_a 2$$ `=` $$0.3010$$ $$\log_a 3$$ `=` $$0.4771$$ `2\log_a 2-2\log_a 3` `=` `2(0.3010)-2(0.4771)` `=` `0.60206-0.95424` `=` `-0.3522` `-0.3522`
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