Logarithmic Equations 2
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Question 1 of 5
1. Question
Solve for `x``log_2 x=log_2 y+4log_2 z`Hint
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$Remove the coefficient from the second term$$\log_{2} x$$ `=` $$\log_{2} y+4\log_{2} z$$ $$\log_{2} x$$ `=` $$\log_{2} y+\log_{2} z^\color{#004ec4}{4}$$ `log_b x^p=p log_b x` Contract the right side$$\log_{2} x$$ `=` $$\log_\color{#9a00c7}{2} \color{#00880A}{y}+\log_\color{#9a00c7}{2} \color{#e65021}{z^4}$$ $$\log_{2} x$$ `=` $$\log_\color{#9a00c7}{2} {\color{#00880A}{y}}{\color{#e65021}{z^4}}$$ `log_b xy=log_b x+log_b y` Since the bases of both sides are the same, the logarithm can be dropped$$\log_{2} \color{#00880A}{x}$$ `=` $$\log_{2} \color{#00880A}{yz^4}$$ $$\color{#00880A}{x}$$ `=` $$\color{#00880A}{yz^4}$$ `x=yz^4` -
Question 2 of 5
2. Question
Solve for `a``log_10 a=3log_10 x-1`Hint
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x}-\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$Remove the coefficient from the second term$$\log_{10} a$$ `=` $$3\log_{10} x-1$$ $$\log_{10} a$$ `=` $$\log_{10} x^\color{#004ec4}{3}-1$$ `log_b x^p=p log_b x` Transform the constant (third term) into a logarithmic term$$\log_{10} a$$ `=` $$\log_{10} x^3-1$$ $$\log_{10} a$$ `=` $$\log_{10} x^3-\log_\color{#9a00c7}{10} \color{#9a00c7}{10}$$ `1=log_{10} 10` Contract the right side$$\log_{10} a$$ `=` $$\log_\color{#9a00c7}{10} \color{#00880A}{x^3}-\log_\color{#9a00c7}{10} \color{#e65021}{10}$$ $$\log_{10} a$$ `=` $$\log_\color{#9a00c7}{10} {\frac{\color{#00880A}{x^3}}{\color{#e65021}{10}}}$$ $$log_b \frac{x}{y}=log_b x-\log_b y$$ Since the bases of both sides are the same, the logarithm can be dropped$$\log_{10} \color{#00880A}{a}$$ `=` $$\log_{10} \color{#00880A}{\frac{x^3}{10}}$$ $$\color{#00880A}{a}$$ `=` $$\color{#00880A}{\frac{x^3}{10}}$$ `a=(x^3)/10` -
Question 3 of 5
3. Question
Solve for `x``log_a x=3log_a y+1`Hint
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$Remove the coefficient from the second term$$\log_{a} x$$ `=` $$3\log_{a} y+1$$ $$\log_{a} x$$ `=` $$\log_{a} y^\color{#004ec4}{3}+1$$ `log_b x^p=p log_b x` Transform the constant (third term) into a logarithmic term$$\log_{a} x$$ `=` $$\log_{a} y^3+1$$ $$\log_{a} x$$ `=` $$\log_{a} y^3+\log_\color{#9a00c7}{a} \color{#9a00c7}{a}$$ `1=log_{a} a` Contract the right side$$\log_{a} x$$ `=` $$\log_\color{#9a00c7}{a} \color{#00880A}{y^3}+\log_\color{#9a00c7}{a} \color{#e65021}{a}$$ $$\log_{a} x$$ `=` $$\log_\color{#9a00c7}{a} {\color{#00880A}{y^3}}{\color{#e65021}{a}}$$ `log_b xy=log_b x+log_b y` Since the bases of both sides are the same, the logarithm can be dropped$$\log_{a} \color{#00880A}{x}$$ `=` $$\log_{a} \color{#00880A}{y^3a}$$ $$\color{#00880A}{x}$$ `=` $$\color{#00880A}{y^3a}$$ `x=y^3a` -
Question 4 of 5
4. Question
Solve for `a``log_10 a=2-log_10 x`Hint
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x}-\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$Add a logarithmic term to the constant (second term)$$\log_{10} a$$ `=` $$2-\log_{10} x$$ $$\log_{10} a$$ `=` $$2\log_\color{#9a00c7}{10} \color{#9a00c7}{10}-\log_{10} x$$ `1=log_{10} 10` Remove the coefficient from the second term$$\log_{10} a$$ `=` $$2\log_{10} 10-\log_{10} x$$ $$\log_{10} a$$ `=` $$\log_{10} 10^\color{#004ec4}{2}-\log_{10} x$$ `log_b x^p=p log_b x` Contract the right side$$\log_{10} a$$ `=` $$\log_\color{#9a00c7}{10} \color{#00880A}{10^2}-\log_\color{#9a00c7}{10} \color{#e65021}{x}$$ $$\log_{10} a$$ `=` $$\log_\color{#9a00c7}{10} {\frac{\color{#00880A}{10^2}}{\color{#e65021}{x}}}$$ $$log_b \frac{x}{y}=log_b x-\log_b y$$ Since the bases of both sides are the same, the logarithm can be dropped$$\log_{10} \color{#00880A}{a}$$ `=` $$\log_{10} \color{#00880A}{\frac{10^2}{x}}$$ $$\color{#00880A}{a}$$ `=` $$\color{#00880A}{\frac{100}{x}}$$ `a=(100)/x` -
Question 5 of 5
5. Question
Solve for `x``log_10 2+2log_10 x-log_10 50=0`- `x=` (5)
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Laws of Logarithms
$$\log_{\color{#9a00c7}{b}} {\color{#00880A}{x}\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x} + \log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_{\color{#9a00c7}{b}} \frac{\color{#00880A}{x}}{\color{#e65021}{y}}=\log_{\color{#9a00c7}{b}} \color{#00880A}{x}-\log_{\color{#9a00c7}{b}} \color{#e65021}{y}$$$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} 1=0$$Remove the coefficient from the second term$$\log_{10} 2+2\log_{10} x-\log_{10} 50$$ `=` $$0$$ $$\log_{10} 2+\log_{10} x^\color{#004ec4}{2}-\log_{10} 50$$ `=` $$0$$ `log_b x^p=p log_b x` Transform the constant (fourth term) into a logarithmic term$$\log_{10} 2+\log_{10} x^2-\log_{10} 50$$ `=` $$0$$ $$\log_{10} 2+\log_{10} x^2-\log_{10} 50$$ `=` $$\log_\color{#9a00c7}{10} 1$$ `0=log_{10} 1` Contract the left side$$\log_\color{#9a00c7}{10} \color{#00880A}{2}+\log_\color{#9a00c7}{10} \color{#e65021}{x^2}-\log_{10} 50$$ `=` $$\log_{10} 1$$ $$\log_\color{#9a00c7}{10} \color{#00880A}{2}\color{#e65021}{x^2}-\log_{10} 50$$ `=` $$\log_{10} 1$$ `log_b xy=log_b x+log_b y` $$\log_\color{#9a00c7}{10} \color{#00880A}{2x^2}-\log_\color{#9a00c7}{10} \color{#e65021}{50}$$ `=` $$\log_{10} 1$$ $$\log_\color{#9a00c7}{10} {\frac{\color{#00880A}{2x^2}}{\color{#e65021}{50}}}$$ `=` $$\log_{10} 1$$ $$log_b \frac{x}{y}=log_b x-\log_b y$$ Since the bases of both sides are the same, the logarithm can be dropped$$\log_{10} \color{#00880A}{\frac{2x^2}{50}}$$ `=` $$\log_{10} \color{#00880A}{1}$$ $$\color{#00880A}{\frac{2x^2}{50}}$$ `=` $$\frac{1}{1}$$ $$2x^2$$ `=` $$50$$ Cross multiply `2x^2``divide2` `=` `50``divide2` Divide both sides by `2` `sqrt(x^2)` `=` `sqrt25` Get the square root of both sides `x` `=` `5` `x=5`
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