Solving Exponential Equations

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GRE Quantitative Reasoning › Solving Exponential Equations

Questions 1 - 10

1

Find one possible value of , given the following equation:

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

Cannot be determined from the information given.

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:

Solve this to get t.

2

Solve for .

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

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

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

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 $\ln (x^y)=y\ln (x)$

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

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

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

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

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

3

Solve for :

$2^{x+1}=8^4$

Explanation

Step 1: Rewrite the right side as a power of :

$8^4=8\cdot8\cdot8\cdot8$

$8^4=2^3\cdot2^3\cdot2^3\cdot2^3$

$8^4=2^{3+3+3+3}=2^{3\cdot 4}=2^{12}$

Step 2: Rewrite the original equation:

$2^{x+1}=2^{12}$

Step 3: Since the bases are equal, I can set the exponents equal.

So, $x+1=12\implies x=11$

4

Solve for .

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

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

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

$x=\ln2+5}$

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

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: $\ln (e)=1$

$4x-5=\ln8$

$4x=\ln8+5$

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

5

Solve this exponential equation for

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

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

$x = 12\ln$

$x = 10 \times \ln (6)$

$x = 10 \times 72\ln$

Explanation

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

Isolate the variable by dividing by 6.

$e ^{0.1x} = 12$

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

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

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

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

6

Find one possible value of , given the following equation:

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

Cannot be determined from the information given.

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:

Solve this to get t.

7

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

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

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

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

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

Explanation

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

To solve, use the natural $\log$ .

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

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

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

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

8

Solve:

Explanation

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..

...

9

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

$x = \log (57) -3$

$x = \log (57) + 3$

$x = \log (57)$

$x = \log (3) -50$

Explanation

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

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.

$x = \log (57) -3$

10

Solve for :

$x=\frac {1}{9}$

Explanation

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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