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High School Math : Solving Exponential Equations

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Example Question #1 : Solving Exponential Equations

Which value for satisfies the equation $4^x = \frac{192}{x}$ ?

Possible Answers:

$\frac{2}{3}$

$\frac{3}{2}$

Correct answer:

Explanation:

is the only choice from those given that satisfies the equation. Substition of for gives:

$4^3 = \frac{192}{3} \rightarrow 64 = 64$

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Example Question #2 : Solving Exponential Equations

Solve for :

2x^2 - 32 =0

Possible Answers:

$x = \{-4, 0, 4\}$

x = 4

x = -4

$x = \{-4, 4\}$

x = 16

Correct answer:

$x = \{-4, 4\}$

Explanation:

To solve for in the equation 2x^2 - 32 =0

Factor out of the expression on the left of the equation:

$\rightarrow 2(x^2 - 16) =0$

Use the "difference of squares" technique to factor the parenthetical term on the left side of the equation.

$\rightarrow 2(x+4)(x-4)=0$

Any variable that causes any one of the parenthetical terms to become will be a valid solution for the equation. x+4 becomes when is , and x-4 becomes when is , so the solutions are and .

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Example Question #3 : Solving Exponential Equations

Solve for (nearest hundredth):

$8^{y} = 100$

Possible Answers:

1.78

2.53

2.21

1.23

1.10

Correct answer:

2.21

Explanation:

Take the common logarithm of both sides and solve for :

$8^{y} = 100$

$\log 8^{y} = \log 100$

$y \log 8 =2$

$y =\frac{2}{\log 8} \approx \frac{2}{0.9031} \approx 2.21$

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Example Question #4 : Solving Exponential Equations

Solve for (nearest hundredth):

$4 ^{2x - 7} = \frac{1}{256}$

Possible Answers:

x=2

x = 5.5

x=-4

x = 1.5 x = 1 .5

x = - 5.5

Correct answer:

x = 1.5 x = 1 .5

Explanation:

$\frac{1}{256} = \frac{1}{4 ^{4}} = 4 ^{-4}$ , so $4 ^{2x - 7} = \frac{1}{256}$ can be rewritten as

$4 ^{2x - 7} = 4 ^{-4}$

$\log_{4} 4 ^{2x - 7} =\log_{4} 4 ^{-4}$

2x - 7 = -4

2x - 7 + 7 = -4+ 7

2x = 3

x = 1 .5

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Example Question #5 : Solving Exponential Equations

Solve for (nearest hundredth):

$80 ^{t} = 39$

Possible Answers:

t=1.20

t=0.62

t =0.84

t=0.72

t=0.31

Correct answer:

t =0.84

Explanation:

One method: Take the natural logarithm of both sides and solve for :

$80 ^{t} = 39$

$\ln 80 ^{t} = \ln 39$

$t \ln 80 = \ln 39$

$t =\frac{ \ln 39}{ \ln 80} \approx \frac{ 3.66}{ 4.38} \approx 0.84$

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Example Question #1 : Solving Exponential Equations

Solve for :

$3 ^{4x + 7} = 9 ^{2x +3}$

Possible Answers:

$x = 1\frac{1}{2}$

x=0

x = 2

x = 1

The equation has no solution.

Correct answer:

The equation has no solution.

Explanation:

Since $9 = 3^{2}$ , we can rewrite this equation by subsituting and applying the power rule:

$3 ^{4x + 7} = 9 ^{2x +3}$

$3 ^{4x + 7} = \left (3 ^{2} \right ) ^{2x +3}$

$3 ^{4x + 7} =3 ^{2 \left ( 2x +3 \right )}$

$3 ^{4x + 7} =3 ^{4x +6}$

$\log_{3} 3 ^{4x + 7} = \log_{3} 3 ^{4x +6}$

4x + 7 = 4x + 6

4x + 7 -4x = 4x + 6-4x

7 = 6

This statement is identically false, which means that the original equation is identically false. There is no solution.

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Example Question #1 : Solving Exponential Equations

Solve for :

$\left (\frac{1}{4} \right) ^{x - 7} = 2^{2x +5}$

Possible Answers:

x = 4.5

The equation has no solution

x = -9.5

x = 2.25

x = -4.75

Correct answer:

x = 2.25

Explanation:

$\frac{1}{4} = \frac{1}{2^{2}} =2^{-2}$ , so we can rewrite the equation as follows:

$\left (\frac{1}{4} \right) ^{x - 7} = 2^{2x +5}$

$\left (2^{-2}) ^{x - 7} = 2^{2x +5}$

$2^{-2 \left ( x - 7 \right )} = 2^{2x +5}$

$2^{-2x +14 \right )} = 2^{2x +5}$

$\log_2 2^{-2x +14 \right )} =\log_2 2^{2x +5}$

-2x + 14 = 2x + 5

-2x + 14 +2x -5= 2x + 5+2x -5

9= 4x

x = 2.25

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Example Question #8 : Solving Exponential Equations

$y=\frac{3x^2-5}{2x^2+7}$

What are the y-intercepts of the equation?

Possible Answers:

y=0

This equation does not have a y-intercept.

$y=\frac{5}{}7$

$y=-\frac{5}{7}$

$y=\frac{6}{35}$

Correct answer:

$y=-\frac{5}{7}$

Explanation:

To find the y-intercepts, set x=0 and solve.

$y=\frac{3x^2-5}{2x^2+7}$

$y=\frac{3(0)^2-5}{2(0)^2+7}$

$y=\frac{-5}{7}$

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Example Question #9 : Solving Exponential Equations

$y=\frac{4x^2-2}{2x^3-27}$

What are the y-intercepts of the equation?

Possible Answers:

$y=\frac{2}{27}$

There are no y-intercepts for this equation.

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

y=0

$y=\frac{27}{2}$

Correct answer:

$y=\frac{2}{27}$

Explanation:

To find the y-intercepts, set x=0 and solve.

$y=\frac{4x^2-2}{2x^3-27}$

$y=\frac{4(0)^2-2}{2(0)^3-27}$

$y=\frac{-2}{-27}$

$y=\frac{2}{27}$

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Example Question #1 : Solving Exponential Equations

$y=\frac{x-3}{x^2-12}$

What are the x-intercepts of this equation?

Possible Answers:

x=3

$x=\frac{1}{4}$

x=4

x=3, -3

$x=2\sqrt{3}$

Correct answer:

x=3

Explanation:

To find the x-intercepts, set the numerator equal to zero.

0=x-3

3=x

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High School Math : Solving Exponential Equations

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : 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

Example Question #1 : Solving Exponential Equations

Example Question #1 : Solving Exponential Equations

Example Question #8 : Solving Exponential Equations

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Example Question #1 : Solving Exponential Equations

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