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Algebra II : Solving Exponential Equations

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Example Question #687 : Exponents

Solve the equation: $e^{6x-4} = \frac{1}{e}$

Possible Answers:

$-\frac{1}{2}$

$\frac{7}{12}$

$-\frac{3}{4}$

$\frac{1}{2}$

$\frac{12}{7}$

Correct answer:

$\frac{1}{2}$

Explanation:

Notice that we can rewrite the right side of the equation as an exponent.

The equation $e^{6x-4} = \frac{1}{e}$ becomes:

$e^{6x-4} = e^{-1}$

Now that the bases of the exponential term are equal, the powers can be set equal to each other.

6x-4 = -1

Add 4 on both sides.

6x-4 +4= -1+4

Simplify the equation.

6x=3

Divide by six on both sides.

$\frac{ 6x}{6}=\frac{3}{6}$

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

The answer is: $\frac{1}{2}$

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Example Question #688 : Exponents

Solve: $10^{2x-3}= 100^{9-3x}$

Possible Answers:

$\frac{11}{8}$

$\frac{18}{5}$

$\frac{17}{3}$

$\frac{23}{12}$

$\frac{21}{8}$

Correct answer:

$\frac{21}{8}$

Explanation:

Change the base on the right side to base ten.

100= 10^2

Replace the hundred with this term.

$10^{2x-3}= 10^{2(9-3x)}$

Set the powers equal to each other now that the bases are equivalent.

2x-3 = 2(9-3x)

Solve for the x-variable. Distribute the two on the right side.

2x-3 = 18-6x

Add on both sides.

2x-3+6x = 18-6x+6x

8x-3 = 18

Add three on both sides.

8x-3+3 = 18+3

8x=21

Divide by eight on both sides.

$\frac{8x}{8}=\frac{21}{8}$

The answer is: $\frac{21}{8}$

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Example Question #689 : Exponents

Solve: $8^{9x} = 8^{-3x+5}$

Possible Answers:

$\frac{1}{3}$

$\frac{5}{12}$

$\frac{1}{4}$

$\frac{5}{3}$

$\frac{3}{5}$

Correct answer:

$\frac{5}{12}$

Explanation:

The bases of the equation are alike, which means we do not need to change the power. We can simply set the powers equal and solve for the unknown variable.

9x=-3x+5

Add on both sides.

9x+3x=-3x+5+3x

12x=5

Divide by 12 on both sides.

$\frac{12x}{12}=\frac{5}{12}$

The answer is: $\frac{5}{12}$

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Example Question #690 : Exponents

Solve: $e^{3x+1} =\frac{1}{e^5}$

Possible Answers:

$\frac{1}{2}$

$-\frac{1}{2}$

Correct answer:

Explanation:

Rewrite the right side with a negative exponent. The goal is to establish similar bases to set the powers equal to each other.

$\frac{1}{e^5} = e^{-5}$

$e^{3x+1} =e^{-5}$

Set the powers equal to each other.

3x+1=-5

Subtract one from both sides.

3x+1-1=-5-1

3x=-6

Divide by three sides.

$\frac{3x}{3}=\frac{-6}{3}$

The answer is:

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Example Question #41 : Solving Exponential Equations

Solve: $10(1000)^{3x+1} = (\frac{1}{100})^{5x}$

Possible Answers:

$-\frac{4}{19}$

$\frac{2}{19}$

$-\frac{2}{13}$

$-\frac{31}{19}$

Correct answer:

$-\frac{4}{19}$

Explanation:

Convert all the terms to base ten.

$(10^1)(10)^{3(3x+1)} = (10^{-2})^{5x}$

Use distribution to simplify the powers in parentheses.

$(10^1)(10)^{9x+3} = 10^{-10x}$

On the left side, since we are multiplying powers of the same base, we can add the exponents.

$10^{1+9x+3}= 10^{-10x}$

Set the powers equal to each other now that the bases are similar.

1+9x+3 = -10x

9x+4 = -10x

Subtract on both sides.

9x+4 -9x= -10x-9x

4=-19x

Divide by negative 19 on both sides.

$\frac{4}{-19}=\frac{-19x}{-19}$

The answer is: $-\frac{4}{19}$

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Example Question #42 : Solving Exponential Equations

Solve: $4^{3x} = 64^{3x-3}$

Possible Answers:

$\frac{3}{2}$

$-\frac{1}{6}$

$-\frac{2}{3}$

$-\frac{3}{2}$

Correct answer:

$\frac{3}{2}$

Explanation:

Convert the base of the right side to base four.

$4^{3x} = 4^{3(3x-3)}$

With common bases, the powers can be set equal to each other.

3x=3(3x-3)

Simplify the right side.

3x=9x-9

Subtract on both sides.

3x-9x=9x-9-9x

-6x =9

Divide by negative six on both sides.

$\frac{-6x }{-6}=\frac{-9}{-6}$

Reduce both fractions.

The answer is: $\frac{3}{2}$

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Example Question #43 : Solving Exponential Equations

Solve: $16^{2x-3} = 64^{6x}$

Possible Answers:

$-\frac{2}{9}$

$-\frac{2}{3}$

$\frac{7}{5}$

$-\frac{3}{11}$

$-\frac{3}{7}$

Correct answer:

$-\frac{3}{7}$

Explanation:

In order to solve this equation, we will need to change the bases of both terms.

Notice that both bases can be four raised to some power.

16=4^2

64=4^3

Replace the terms.

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

Set the powers equal to each other.

2(2x-3)=18x

Divide by two on both sides.

$\frac{2(2x-3)}{2}=\frac{18x}{2}$

2x-3=9x

Subtract on both sides.

2x-3-2x=9x-2x

-3=7x

Divide by seven on both sides.

$\frac{-3}{7}=\frac{7x}{7}$

The answer is: $-\frac{3}{7}$

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Example Question #44 : Solving Exponential Equations

Solve for .

$5^{2x+5}=5^{17}$

Possible Answers:

Correct answer:

Explanation:

When solving exponential equations, we need to ensure that we have the same base. When that happens, our equations our based on the exponents.

$5^{2x+5}=5^{17}$ With same base, we can write:

2x+5=17 Subtract on both sides.

2x=12 Divide on both sides.

x=6

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Example Question #45 : Solving Exponential Equations

Solve for .

$3^{4x-23}=3^{17}$

Possible Answers:

Correct answer:

Explanation:

When solving exponential equations, we need to ensure that we have the same base. When that happens, our equations our based on the exponents.

$3^{4x-23}=3^{17}$ With the same base, we can write:

4x-23=17 Add on both sides.

4x=40 Divide on both sides.

x=10

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Example Question #46 : Solving Exponential Equations

Solve: $2^{15x} = 8^{90x}$

Possible Answers:

$\textup{No solution.}$

Correct answer:

Explanation:

Rewrite the second term in terms of base two.

8=2^3

Rewrite the equation.

$2^{15x} = 2^{3(90x)}$

Set the powers equal to each other.

15x = 270x

Do NOT divide by x on both sides or we will get no solution!

Instead, subtract 15x from both sides.

The equation becomes: 0=255x

Divide by 255 on both sides.

The answer is:

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Algebra II : Solving Exponential Equations

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #687 : Exponents

Example Question #688 : Exponents

Example Question #689 : Exponents

Example Question #690 : Exponents

Example Question #41 : Solving Exponential Equations

Example Question #42 : Solving Exponential Equations

Example Question #43 : Solving Exponential Equations

Example Question #44 : Solving Exponential Equations

Example Question #45 : Solving Exponential Equations

Example Question #46 : Solving Exponential Equations

All Algebra II Resources

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