Logarithmic Equations 1
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Question 1 of 4
1. Question
Solve for `N``log_5 N=-3`- `N=` (1/125)
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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$$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$$ `=` $$x$$ $$\log_{\color{#9a00c7}{5}} \color{#00880a}{N}$$ `=` $$-3$$ `N` `=` `N` `a` `=` `5` `x` `=` `-3` Substitute the components into the exponent form$$\color{#00880a}{N}$$ `=` $${\color{#9a00c7}{a}}^x$$ $$\color{#00880a}{N}$$ `=` $${\color{#9a00c7}{5}}^{-3}$$ Solve for the value of `N``N` `=` `5^(-3)` `N` `=` `1/(5^3)` Reciprocate `5^(-3)` `N` `=` `1/125` `N=1/125` -
Question 2 of 4
2. Question
Solve for `N``log_3 N=-4`- `N=` (1/81)
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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$$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$$ `=` $$x$$ $$\log_{\color{#9a00c7}{3}} \color{#00880a}{N}$$ `=` $$-4$$ `N` `=` `N` `a` `=` `3` `x` `=` `-4` Substitute the components into the exponent form$$\color{#00880a}{N}$$ `=` $${\color{#9a00c7}{a}}^x$$ $$\color{#00880a}{N}$$ `=` $${\color{#9a00c7}{3}}^{-4}$$ Solve for the value of `N``N` `=` `3^(-4)` `N` `=` `1/(3^4)` Reciprocate `3^(-4)` `N` `=` `1/81` `N=1/81` -
Question 3 of 4
3. Question
Solve for `a``log_a 27=3`- `a=` (3)
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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$$\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$$ `=` $$x$$ $$\log_{\color{#9a00c7}{a}} \color{#00880a}{27}$$ `=` $$3$$ `N` `=` `27` `a` `=` `a` `x` `=` `3` Substitute the components into the exponent form$$\color{#00880a}{N}$$ `=` $${\color{#9a00c7}{a}}^x$$ $$\color{#00880a}{27}$$ `=` $${\color{#9a00c7}{a}}^{3}$$ Solve for the value of `a``27` `=` `a^3` `root (3)(27)` `=` `root (3)(a^3)` Find the cube root of both sides `3` `=` `a` `a` `=` `3` `a=3` -
Question 4 of 4
4. Question
Solve for `x``log_b x=3log_b 2+log_b 4`- `x=` (32)
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$Remove the coefficient from the second term$$\log_{b} x$$ `=` $$3\log_{b} 2+\log_{b} 4$$ $$\log_{b} x$$ `=` $$\log_{b} 2^\color{#004ec4}{3}+\log_{b} 4$$ `log_b x^p=p log_b x` $$\log_{b} x$$ `=` $$\log_{b} 8+\log_{b} 4$$ Contract the right side$$\log_{b} x$$ `=` $$\log_\color{#9a00c7}{b} \color{#00880A}{8}+\log_\color{#9a00c7}{b} \color{#e65021}{4}$$ $$\log_{b} x$$ `=` $$\log_\color{#9a00c7}{b} ({\color{#00880A}{8}})({\color{#e65021}{4}})$$ `log_b xy=log_b x+log_b y` $$\log_{b} x$$ `=` $$\log_b 32$$ Since the bases of both sides are the same, the logarithm can be dropped$$\log_{b} \color{#00880A}{x}$$ `=` $$\log_{b} \color{#00880A}{32}$$ $$\color{#00880A}{x}$$ `=` $$\color{#00880A}{32}$$ `x=32`
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