Solving Exponential Equations
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Question 1 of 5
1. Question
Solve for `x` using Change of Base`3^(x+1)=8`Round answer to `4` decimal places- `x=` (0.8928)
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Exponent Form
$$\color{#00880a}{N}={\color{#9a00c7}{a}}^x$$Logarithmic Form
$$x=\log_{\color{#9a00c7}{a}} \color{#00880a}{N}$$Change of Base Formula
$$\log_\color{#9a00c7}{a} \color{#00880A}{N}=\frac{\log_b \color{#00880A}{N}}{\log_b \color{#9a00c7}{a}}$$Transform the given exponential equation to logarithmic form$$\color{#9a00c7}{3}^{x+1}$$ `=` $$\color{#00880a}{8}$$ $$x+1$$ `=` $$\log_\color{#9a00c7}{3} \color{#00880a}{8}$$ Use the change of base formula, then use the calculator to solve the logarithm$$x+1$$ `=` $$\log_\color{#9a00c7}{3} \color{#00880a}{8}$$ $$x+1$$ `=` $$\frac{\log_{10} \color{#00880a}{8}}{\log_{10} \color{#9a00c7}{3}}$$ Calculators use `10` as base for the log function $$x+1$$ `=` $$1.8928$$ Compute using the calculator `x+1` `-1` `=` `1.8928` `-1` Subtract `1` from both sides `x` `=` `0.8928` `x=0.8928` -
Question 2 of 5
2. Question
Solve for `x``2^x=5^(x-1)`Round answer to `4` decimal places- `x=` (1.7565)
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Laws of Logarithms
$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$Insert logarithms of the same base to both sides, then solve for `x`$$\color{#D800AD}{2^x}$$ `=` $$\color{#D800AD}{5^{x-1}}$$ $$\log_{10} \color{#D800AD}{2^x}$$ `=` $$\log_{10} \color{#D800AD}{5^{x-1}}$$ Calculators use base `10` for log function $$\color{#004ec4}{x}\log_{10} 2$$ `=` $$\color{#004ec4}{(x-1)}\log_{10} 5$$ `log_b x^p=p log_b x` `xlog_(10) 2` `=` `x log_(10) 5-log_(10) 5` Distribute `xlog_(10) 2` `+log_(10) 5` `=` `x log_(10) 5-log_(10) 5` `+log_(10) 5` Add `log_(10) 5` to both sides `xlog_(10) 2+log_(10) 5` `=` `x log_(10) 5` `xlog_(10) 2+log_(10) 5` `-xlog_(10) 2` `=` `x log_(10) 5` `-xlog_(10) 2` Subtract `xlog_(10) 2` from both sides `log_(10) 5` `=` `x(log_(10) 5-log_(10) 2)` Factorize $$\frac{\log_{10} 5}{\color{#CC0000}{\log_{10} 5-\log_{10} 2}}$$ `=` $$\frac{x(\log_{10} 5-\log_{10} 2)}{\color{#CC0000}{\log_{10} 5-\log_{10} 2}}$$ Divide both sides by `log_(10) 5-log_(10) 2` `(log_(10) 5)/(log_(10) 5-log_(10) 2)` `=` `x` `x` `=` `1.7565` Compute using the calculator `x=1.7565` -
Question 3 of 5
3. Question
Solve for `x``10^(3x-2)=5^(4x)`Round answer to `3` decimal places- `x=` (9.798)
Hint
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Laws of Logarithms
$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$Insert logarithms of the same base to both sides, then solve for `x`$$\color{#D800AD}{10^{3x-2}}$$ `=` $$\color{#D800AD}{5^{4x}}$$ $$\log_{10} \color{#D800AD}{10^{3x-2}}$$ `=` $$\log_{10} \color{#D800AD}{5^{4x}}$$ Calculators use base `10` for log function $$\color{#004ec4}{(3x-2)}\log_{10} 10$$ `=` $$\color{#004ec4}{(4x)}\log_{10} 5$$ `log_b x^p=p log_b x` $$(3x-2)(\color{#9a00c7}{1})$$ `=` $$\color{#004ec4}{(4x)}\log_{10} 5$$ `log_b b=1` `3x-2` `+2` `=` `4x log_(10) 5` `+2` Add `2` to both sides `3x` `=` `4x log_(10) 5+2` `3x` `-4xlog_(10) 5` `=` `4x log_(10) 5+2` `-4xlog_(10) 5` Subtract `4xlog_(10) 5` from both sides `x(3-4log_(10) 5)` `=` `2` Factorize $$\frac{x(3-4\log_{10} 5)}{\color{#CC0000}{3-4\log_{10} 5}}$$ `=` $$\frac{2}{\color{#CC0000}{3-4\log_{10} 5}}$$ Divide both sides by `3-4\log_(10) 5` `x` `=` `2/(3-4\log_(10) 5)` `x` `=` `9.798` Compute using the calculator `x=9.798` -
Question 4 of 5
4. Question
Solve for `x``6^x=0.00025`Round answer to `3` decimal places- `x=` (-4.629)
Hint
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Fantastic!
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Laws of Logarithms
$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$Transform the decimal to exponential form$$6^x$$ `=` $$0.00025$$ $$6^x$$ `=` $$\frac{25}{100 000}$$ `0.00025=25/(100 000)` $$6^x$$ `=` $$\frac{1}{4000}$$ Simplify $$6^x$$ `=` $$4000^{-1}$$ Reciprocate `1/(4000)` Insert logarithms of the same base to both sides, then solve for `x`$$\color{#D800AD}{6^{x}}$$ `=` $$\color{#D800AD}{4000^{-1}}$$ $$\log_{10} \color{#D800AD}{6^{x}}$$ `=` $$\log_{10} \color{#D800AD}{4000^{-1}}$$ Calculators use `10` as base for the log function $$\color{#004ec4}{x}\log_{10} 6$$ `=` $$\color{#004ec4}{(-1)}\log_{10} {4000}$$ `log_b x^p=p log_b x` $$\frac{x\log_{10} 6}{\color{#CC0000}{\log_{10} 6}}$$ `=` $$\frac{(-1)\log_{10} 4000}{\color{#CC0000}{\log_{10} 6}}$$ Divide both sides by `log_(10) 6` `x` `=` `((-1)log_(10) 4000)/(log_(10) 6)` `x` `=` `-4.629` Compute using calculator `x=-4.629` -
Question 5 of 5
5. Question
Solve for `x``(3/5)^x=10^(-5)`Round answer to `4` decimal places- `x=` (-22.5379)
Hint
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Excellent!
Incorrect
Laws of Logarithms
$$\log_b x^\color{#004ec4}{p}=\color{#004ec4}{p}\log_b x$$$$\log_{\color{#9a00c7}{b}} \color{#9a00c7}{b}=1$$Insert logarithms of the same base to both sides, then solve for `x`$$\color{#D800AD}{\frac{3}{5}^{x}}$$ `=` $$\color{#D800AD}{10^{-5}}$$ $$\log_{10} \color{#D800AD}{\frac{3}{5}^{x}}$$ `=` $$\log_{10} \color{#D800AD}{10^{-5}}$$ Calculators use `10` as base for the log function $$\color{#004ec4}{x}\log_{10} \frac{3}{5}$$ `=` $$\color{#004ec4}{(-5)}\log_{10} 10$$ `log_b x^p=p log_b x` $$x\log_{10} \frac{3}{5}$$ `=` $$(-5)(\color{#9a00c7}{1})$$ `log_b b=1` $$\frac{x\log_{10} \frac{3}{5}}{\color{#CC0000}{\log_{10} \frac{3}{5}}}$$ `=` $$\frac{-5}{\color{#CC0000}{\log_{10} \frac{3}{5}}}$$ Divide both sides by `\log_(10) (3/5)` `x` `=` `-5/(\log_(10) (3/5))` `x` `=` `-22.5379` Compute using the calculator `x=-22.5379`
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