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Precalculus : Use Logarithms To Solve Exponential Equations and Inequalities

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Example Questions

Example Question #1 : Use Logarithms To Solve Exponential Equations And Inequalities

Solving an exponential equation.

Solve for ,

$5^{x}=250$ .

Possible Answers:

$x=\frac{3}{2}$

x=5

x=3+log2

x=30

$x=3+log_{5}2$

Correct answer:

$x=3+log_{5}2$

Explanation:

We recall the property:

$log_b(x)=y \leftrightarrow b^y=x$

b=5, y=250

log_5(250)=x

$x=log_{5}(125\cdot 2)=log_{5}125+log_{5}2$

Now, $log_{5}125=log_{5}(5^{3})=3$ .

Thus

$x=3+log_{5}2$ .

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Example Question #2 : Use Logarithms To Solve Exponential Equations And Inequalities

Solving an exponential equation.

Solve

$10^{3x}=100$

Possible Answers:

$x=\frac{3}{2}$

$x=\frac{2}{3}$

x=3

x=2

x=10

Correct answer:

$x=\frac{2}{3}$

Explanation:

Use log (which is just $log_{10}$ , by convention) to solve.

$log(10^{3x})=log(100)$

$log(10^{3x})=log(10^2)$

3x=2

$x=\frac{2}{3}$ .

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Example Question #1 : Use Logarithms To Solve Exponential Equations And Inequalities

Solve the equation for using the rules of logarithms.

$\ln(17x)+\ln\left(\frac{x}{3}\right)+\log_3(x^2)=4$

Possible Answers:

$x\approx 1.04$

$x\approx 75.15$

x=3

$x\approx1.86$

x=9

Correct answer:

$x\approx 1.04$

Explanation:

Expanding the logarithms into sums of logarithms will cancel out the first two x terms, resulting in the equation:

$\ln(17)+ln(3)+2\log_3(x)=4$

Combining the first and second terms, then subtracting the new term over will allow you to isolate the variable term.

Divide both sides of the equation by 2, then exponentiate with 3.

Evaluating this term numerically will give the correct answer.

$2\log_3(x)=4-\ln(51)\implies \log_3(x)=2-\frac{\ln(51)}{2}\\ \\ \implies x=3^{2-\ln(51)/2}\approx1.04$

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Example Question #2 : Use Logarithms To Solve Exponential Equations And Inequalities

Solve the following equation:

10^x=175

Possible Answers:

$x\approx22.4$

$x\approx2.24$

x=2.24

$x\approx4.48$

$x\approx0.224$

Correct answer:

$x\approx2.24$

Explanation:

To solve this equation, recall the following property:

log_by=x Can be rewritten as b^x=y

10^x=175

$log_{10}175=x$

Evaluate with your calculator to get

$x\approx2.24$

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Example Question #1 : Exponential Equations And Inequalities

Solve

$\frac{e^{x^2-2}}{e^{3x}}=e^{2x-6}$ .

Possible Answers:

x=1

x=1$ or $4

x=4

x=0$ or 1

x=7$ or $4

Correct answer:

x=1$ or $4

Explanation:

After using the division rule to simplify the left hand side you can take the natural log of both sides.

If you then combine like terms you get a quadratic equation which factors to,

(x-1)(x-4)=0 .

Setting each binomial equal to zero and solving for we get the solution to be x=1$ or $4 .

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Example Question #1 : Use Logarithms To Solve Exponential Equations And Inequalities

Solve for x: 3^x=243

Possible Answers:

x=3

x=-1

x=5

x=1

x=4

Correct answer:

x=4

Explanation:

$3^x=3^4\rightarrow x=4$

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Example Question #6 : Exponential Equations And Inequalities

Solve for x in the following equation: $3^{5x-3}=9^{2x}$

Possible Answers:

x=3

x=5

x=1

x=0

x=-1

Correct answer:

x=3

Explanation:

$3^{5x-3}=3^{(2)(2x)}\rightarrow 5x-3=4x\rightarrow x=3$

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Example Question #2 : Use Logarithms To Solve Exponential Equations And Inequalities

Solve for x using the rules of logarithms: $\log_{3}2x-\log_{3}(x-3)=1$

Possible Answers:

x=2

x=9

x=3

x=4

x=1

Correct answer:

x=9

Explanation:

$log_3 \frac{2x}{x-3}=1\rightarrow 3^{log_3 \frac{2x}{x-3}}=3^1\rightarrow \frac{2x}{x-3}=3\rightarrow x=9$

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Example Question #8 : Exponential Equations And Inequalities

Solve for x: $\log_{4}(3x-2)=2$

Possible Answers:

x=6

x=4

x=3

x=2

x=5

Correct answer:

x=6

Explanation:

$log_4 (3x-2)=2\rightarrow 3x-2=4^2\rightarrow 3x=18\rightarrow x=6$

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Example Question #3 : Use Logarithms To Solve Exponential Equations And Inequalities

Simplify the log expression: logx*log2

Possible Answers:

2log2x

Cannot be simplified any further

log2x

log(x)^2

2logx

Correct answer:

Cannot be simplified any further

Explanation:

The logarithmic expression is as simplified as can be.

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Precalculus : Use Logarithms To Solve Exponential Equations and Inequalities

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #2 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #1 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #2 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #1 : Exponential Equations And Inequalities

Example Question #1 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #6 : Exponential Equations And Inequalities

Example Question #2 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #8 : Exponential Equations And Inequalities

Example Question #3 : Use Logarithms To Solve Exponential Equations And Inequalities

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