Exponential Equations and Inequalities - Pre-Calculus

Card 0 of 40

Question

Solving an exponential equation.

Solve for ,

$5^{x}=250$ .

Answer

We recall the property:

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

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

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

Thus

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

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Question

Solving an exponential equation.

Solve

$10^{3x}=100$

Answer

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

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

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

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

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Question

Solve

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

Answer

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,

.

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

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Question

Solve the equation for using the rules of logarithms.

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

Answer

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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Question

Solve the following equation:

Answer

To solve this equation, recall the following property:

Can be rewritten as

$log_{10}175=x$

Evaluate with your calculator to get

$x\approx2.24$

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Question

Solve for x:

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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

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Question

Simplify the log expression:

Answer

The logarithmic expression is as simplified as can be.

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Question

Solving an exponential equation.

Solve for ,

$5^{x}=250$ .

Answer

We recall the property:

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

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

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

Thus

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

Compare your answer with the correct one above

Question

Solving an exponential equation.

Solve

$10^{3x}=100$

Answer

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

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

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

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

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Question

Solve

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

Answer

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,

.

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

Compare your answer with the correct one above

Question

Solve the equation for using the rules of logarithms.

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

Answer

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.

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

Compare your answer with the correct one above

Question

Solve the following equation:

Answer

To solve this equation, recall the following property:

Can be rewritten as

$log_{10}175=x$

Evaluate with your calculator to get

$x\approx2.24$

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Question

Solve for x:

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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

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Question

Simplify the log expression:

Answer

The logarithmic expression is as simplified as can be.

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