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GRE Subject Test: Math : Solving Exponential Equations

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Example Questions

Example Question #1 : Solving Exponential Equations

Find one possible value of , given the following equation:

$343=7^{15t-12}$

Possible Answers:

t=5

t=0

t=1

t=-3

Cannot be determined from the information given.

Correct answer:

t=1

Explanation:

We begin with the following:

$343=7^{15t-12}$

This can be rewritten as

$7^3=7^{15t-12}$

Recall that if you have two exponents with equal bases, you can simply set the exponents equal to eachother. Do so to get the following:

3=15t-12

Solve this to get t.

15=15t

t=1

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Example Question #2 : Solving Exponential Equations

Solve for .

$4^{2x}=16^{2x-6}$

Possible Answers:

$x=log \ 2$

x=-6

x=4

x=6

Correct answer:

x=6

Explanation:

We need to make the bases equal before attempting to solve for . Since 16=4^2 we can rewrite our equation as

$4^{2x}=4^{2^{2x-6}}$

Remember: the exponent rule $(x^n)^m=x^{n\cdot m}$

$4^{2x}=4^{2(2x-6)}$

Now that our bases are equal, we can set the exponents equal to each other and solve for .

2x=2(2x-6)

2x=4x-12

12=2x

x=6

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Example Question #3 : Solving Exponential Equations

Solve for .

$7^{2y}-3^{y+2}=0$

Possible Answers:

$y=\frac{2\ln3}{2 \ln 7-\ln3}$

y=-1

$y=\frac{2\ln3}{2 \ln 7+\ln3}$

y=1

Correct answer:

$y=\frac{2\ln3}{2 \ln 7-\ln3}$

Explanation:

The first step is to make sure we don't have a zero on one side which we can easily take care of:

$7^{2y}=3^{y+2}$

Now we can take the logarithm of both sides using natural log:

$\ln 7^{2y}=\ln 3^{y+2}$

Note: we can apply the Power Rule here $\ln (x^y)=y\ln (x)$

$2y \ln 7=(y+2)\ln3$

$2y \ln 7=y\ln3+2\ln3$

$2y \ln 7-y\ln3=2\ln3$

$y(2 \ln 7-\ln3)=2\ln3$

$y=\frac{2\ln3}{2 \ln 7-\ln3}$

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Example Question #324 : Gre Subject Test: Math

Solve for .

$3e^{(4x-5)}=24$

Possible Answers:

$x=\frac{\ln8+5}{4}$

$x=\frac{\ln21+5}{4}$

$x=\frac{\ln8-5}{4}$

$x=\ln2+5}$

Correct answer:

$x=\frac{\ln8+5}{4}$

Explanation:

Before beginning to solve for , we need to have a coefficient of :

$e^{(4x-5)}=8$

Now we can take the natural log of both sides:

$\ln e^{(4x-5)}=\ln8$

Note: $\ln (e)=1$

$4x-5=\ln8$

$4x=\ln8+5$

$x=\frac{\ln8+5}{4}$

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Example Question #1 : Solving Exponential Equations

$6^{3}\times6^{n} =6^{12}$

Possible Answers:

n =9

n=4

n =15

n=36

Correct answer:

n =9

Explanation:

$6^{3} \times 6^{n} =6^{12}$

Since the base is for both, then:

3 + n =12 When the base is the same, and you are multiplying, the exponents are added.

n= 9

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Example Question #3 : Solving Exponential Equations

$10^{3x} = 63$

Possible Answers:

x = log (63)

$x = \frac{log (3)}{63}$

x = 63/3

$x = \frac{log(63)}{3}$

Correct answer:

$x = \frac{log(63)}{3}$

Explanation:

$10^{3x} = 63$

To solve, use common $\log$

$\log (10^{3x}) = \log (63)$

$3x\log (10) = \log (63)$

$3x (1) = \log (63)$

$3x = \log (63)$

$x = \frac{\log (63)}{3}$

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Example Question #2 : Solving Exponential Equations

$2^{x} = 40$

Possible Answers:

$x = \ln (40)$

$x = \frac{\\ln (2) }{\\ ln(40) }$

$x =\frac{\\ln (20)}{\\ln (2)}$

$x = \frac{\\ln ( 40)}{\\ln (2)}$

Correct answer:

$x = \frac{\\ln ( 40)}{\\ln (2)}$

Explanation:

$2^{x} = 40$

To solve, use the natural log.

$ln (2^{x}) = ln (40)$

xln (2) = ln(40)

To isolate the variable, divide both sides by ln (2) .

$x = \frac{\\ln (40)}{\\ln (2)}$

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Example Question #3 : Solving Exponential Equations

$4^{x} = 200$

Possible Answers:

$x =\frac{\\ln (200)}{\\ln (4) }$

$x = \frac{\\ln (4)}{\\ln (200) }$

$x =\frac{1}{5\\ln }$

x = 50

Correct answer:

$x =\frac{\\ln (200)}{\\ln (4) }$

Explanation:

$4^{x} = 200$

To solve, use the natural log.

$\\ln (4^{x}) = \ln(200)$

$x\ln (4) = \ln (200)$

$x = \ln (200)$

$x = \frac{ln(200)}{ln (4)}$

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Example Question #4 : Solving Exponential Equations

Solve the equation. Express the solution as a logarithm in base-10.

$-2\times 10^{3x} = - 500$

Possible Answers:

$x = \log (250)$

$x = \frac{\log 3}{250}$

$x=\frac{\log (250)}{3}$

$x =\frac{\log500 }{3}$

Correct answer:

$x=\frac{\log (250)}{3}$

Explanation:

$-2 (10^{3x}) = -500$

Isolate the exponential part of the equation.

$\frac{-2 (10^{3x})}{-2} = \frac{-500}{-2}$

$10^{3x} = 250$

Convert to log form and solve.

$\log_{10} (250) = 3x$

$\log_{10} (250)$ can also be written as $\log 250$ .

$\frac{\log (250) }{3} = x$

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Example Question #5 : Solving Exponential Equations

$8\times e^{0.3x} = 12$

Possible Answers:

$\frac{\\ln (8)}{0.3} = x$

$\frac{\\ln (12)}{0.3} = x$

$\frac{\\ln 0.3}{12} = x$

$\frac{\\ln (1.5)}{0.3} = x$

Correct answer:

$\frac{\\ln (1.5)}{0.3} = x$

Explanation:

$8\times e^{0.3x} = 12$

Simply the exponential part of the equation by dividing both sides by

$e^{0.3x} = \frac{12}{8}$

$e^{0.3x} = 1.5$

Write in logarithm form.

$\log_{e} (1.5) = 0.3x$

Because $\log_{e} (1.5)$ is also written as $\\ln 1.5$

$\frac{\\ln 1.5}{0.3} = x$

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GRE Subject Test: Math : Solving Exponential Equations

Study concepts, example questions & explanations for GRE Subject Test: Math

All GRE Subject Test: Math Resources

Example Questions

Example Question #1 : Solving Exponential Equations

Example Question #2 : Solving Exponential Equations

Example Question #3 : Solving Exponential Equations

Example Question #324 : Gre Subject Test: Math

Example Question #1 : Solving Exponential Equations

Example Question #3 : Solving Exponential Equations

Example Question #2 : Solving Exponential Equations

Example Question #3 : Solving Exponential Equations

Example Question #4 : Solving Exponential Equations

Example Question #5 : Solving Exponential Equations

All GRE Subject Test: Math Resources

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