Solving Exponential Equations - GRE Subject Test: Math

Card 0 of 20

Question

Find one possible value of , given the following equation:

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

Answer

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:

Solve this to get t.

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Question

Solve for .

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

Answer

We need to make the bases equal before attempting to solve for . Since 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 .

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Question

Solve for .

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

Answer

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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Question

Solve for .

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

Answer

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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Question

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

Answer

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

Since the base is for both, then:

When the base is the same, and you are multiplying, the exponents are added.

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Question

$2^{x} = 40$

Answer

$2^{x} = 40$

To solve, use the natural log.

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

To isolate the variable, divide both sides by .

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

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Question

$4^{x} = 200$

Answer

$4^{x} = 200$

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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Question

$10^{3x} = 63$

Answer

$10^{3x} = 63$

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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Question

$2 (2^{x+3}) =355$

Answer

$2 (2^{x+3}) =355$

To solve, use the natural $\log$ .

$\ln (2^{x+3}) =\ln (355/2)$

$(x+3) (\ln )(2) = \ln \frac{355}{2}$

$x + 3= \frac{\ln \frac{355}{2}}{\ln (2)}$

$x = \frac{\ln \frac{355}{2}}{\ln (2)} - 3$

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Question

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

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

Answer

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

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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Question

$6 \times 3^{2y} = 216$

Answer

$6 \times 3^{2y} = 216$

Divide both sides by

$3^{2y} = 36$

Write in logarithm form and solve for

$2y =\log_{3} (36)$

Divide both sides by

$y = \frac{\log_{e} (36)}{2}$

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Question

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

Answer

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

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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Question

$10 \times 3^{\frac{3t}{4}} = 200$

Answer

$10 \times 3^{\frac{3t}{4}} = 200$

Isolate the variable by dividing both sides of the equation by

$3^{\frac{3t}{4}} = 20$

Write in logarithm form.

$\log_{3}= (20)\frac{3^{t}}{4}$

$\frac{4}{3}\log_{3} (20) = t$

Because the solution is in base-3 log, it can be changed to base -10 by using:

$\log_{b} (a) = \frac{\log_{x}(a)}{\log_{x}(b)}$

$\frac{4}{3}\log_{3} (20) = t$

$\frac{4}{3} \times \frac{\\log {20}}{\log{3}} = t$

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Question

Solve this exponential equation for

$6\times e^{0.1x} = 72$

Answer

$6\times e ^{0.1x} = 72$

Isolate the variable by dividing by 6.

$e ^{0.1x} = 12$

$\log_{e} (12) =0.1x$

$\log_{e} (12)$ is the same as $\ln (12)$ .

$\frac{\ln(12)}{0.1} = x$

$x = 10 \times \ln (12)$

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Question

$10^{x+3} - 7 = 50$

Answer

$10^{^{x+3}} - 7 = 50$

Isolate by adding to both sides of the exponential equation.

$10^{^{x+3}} = 57$

Take the common log.

$\log (10^{x+3}) = \log (57)$

Use logarithmic rule 3. An exponent inside a log can be moved outside as a multiplier.

$(x+3) \log (10) = \log (57)$

Simplify. Because $\log (10) = 1$

$(x + 3) = \log (57)$

Isolate the variable by subtracting from both sides.

$x = \log (57) -3$

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Question

$\frac{3000}{2 + 7^{2x}} = 5$

Answer

$\frac{3000}{2 + 7^{2x}} = 5$

$3000 = 5 (7^{2x} +2)$

Simplify by dividing both sides by

$600 = 7^{2x} +2$

Subtract from both sides of the exponential equation.

$598 = 7^{2x}$

Since base is 7, take log 7 of both sides.

$\log_{7} (7^{2x}) =\log_{7} (598)$

Use logarithmic rule 3. An exponent on everything inside a log can be moved out front as a multiplier,

$2x (\log_{7})7 = \log_{7} (598)$

$\log_{7} \times 7 =1$

$2x = \log_{7} (598)$

Simplify by dividing both sides of the exponential equation by 2.

$x = \frac{\log_{7} (598)}{2}$

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Question

Solve for x: $2^{x+1}=128$

Answer

Step 1: Rewrite the right side of the equation into a power of 2.

$128=2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2=2^7$

$2^{x+1}=2^7$

Step 2: We have the same base, so we can equate the exponents.

Step 3: Solve for x. We will subtract 1 from both sides to isolate x.

$\rightarrow x=6$

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Question

Solve for :

$2^{x^2-3x}=4^{5}$

Answer

Step 1: Rewrite as .

$4^5 \equiv 2^{10}$

Step 2: Re-write the equation:

$2^{x^2-3x}=2^{10}$

Step 3: By laws of exponents, if the bases are the same, we can equate the exponents...

We will get

Step 4: Move 10 over and begin factoring:

Step 5: is a correct answer... we can plug it in and see:

$2^{(-2)^2-3(-2)}=2^{4--6}=2^{10}$

Step 6: is the other correct answer...

$2^{(5)^2-3(5)}=2^{25-15}=2^{10}$

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Question

Solve:

Answer

Step 1: Rewrite the right hand side of the equation as a power of 2.

$4^3\equiv 2^6$ . To get this, divide the base by 2 and multiply that 2 to the exponent...

Step 2: Equate the left and right side together

We have the same base, so we equate the exponents together..

...

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Question

Solve for :

Answer

Step 1: Write as ...

$8\cdot8\cdot8=(2\cdot 2\cdot 2)(2\cdot 2\cdot 2)(2\cdot 2\cdot 2)=2^9$

Step 2: Rewrite as in the original equation..

Step 3: By a rule of exponents, I can set the exponents equal if the bases of both exponents are the same...

So,

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