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Algebra II : Solving Exponential Equations

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Example Questions

Example Question #1 : Solving Exponential Equations

Solve the equation for .

$\small 9^x=3^6$

Possible Answers:

$\small x=1$

$\small x=2$

$\small x=0$

$\small x=3$

Correct answer:

$\small x=3$

Explanation:

Begin by recognizing that both sides of the equation have a root term of .

$\small 9^x=3^6$

(3^2)^x=3^6

Using the power rule, we can set the exponents equal to each other.

$3^{(2*x)}=3^6$

$\small 2x=6$

$\small x=3$

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Example Question #1 : Solving Exponential Equations

Solve the equation for .

$\small 3^{2x}=81$

Possible Answers:

x=9

x=1

x=2

x=4

x=3

Correct answer:

x=2

Explanation:

Begin by recognizing that both sides of the equation have the same root term, .

$\small 3^{2x}=81$

$\small 3^{2x}=9^2$

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

We can use the power rule to combine exponents.

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

Set the exponents equal to each other.

2x=4

$\small x=2$

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Example Question #1 : Solving Exponential Equations

In 2009, the population of fish in a pond was 1,034. In 2013, it was 1,711.

Write an exponential growth function of the form $y=ab^{x}}$ that could be used to model , the population of fish, in terms of , the number of years since 2009.

Possible Answers:

y=1034(1.1342)^x

y=1.35(1.0011)^x

y=1.2(1711)^x

y=1711(1.3932)^x

y=1.1(1034)^x

Correct answer:

y=1034(1.1342)^x

Explanation:

Solve for the values of a and b:

In 2009, y=1034 and x=0 (zero years since 2009). Plug this into the exponential equation form:

$1034=ab^{(0)}$ . Solve for to get .

In 2013, y=1711 and x=4 . Therefore,

1711=ab^4 or 1711=(1034)b^4 . Solve for to get

$b=\sqrt[4]{\frac{1711}{1034}}\approx1.1342$ .

Then the exponential growth function is

$y=1034(1.1342)^{x}$ .

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Example Question #1 : Solving Exponential Functions

$4^{5x}=8^{4x-1}$

Solve for .

Possible Answers:

1.5

Correct answer:

1.5

Explanation:

8 and 4 are both powers of 2.

$4^{5x}=8^{4x-1}$

$2^{10x}=2^{12x-3}$

10x=12x-3 -2x=-3 x=1.5

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Example Question #2 : Solving Exponential Functions

Solve for :

$80^{2x+3}=80^{5x-9}$

Possible Answers:

x=80

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

x=2

x=4

No solution

Correct answer:

x=4

Explanation:

Because both sides of the equation have the same base, set the terms equal to each other.

2x+3=5x-9

Add 9 to both sides: 2x+12=5x

Then, subtract 2x from both sides: 12=3x

Finally, divide both sides by 3: x=4

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Example Question #11 : Solving Functions

Solve for :

$25^{-x+5}=125^{6x-10}$

Possible Answers:

$x=\frac{15}{7}$

x=5

No solution

x=2

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

Correct answer:

x=2

Explanation:

125 and 25 are both powers of 5.

Therefore, the equation can be rewritten as

$(5^3)^{6x-10}=(5^2)^{-x+5}$ .

Using the Distributive Property,

$5^{18x-30}=5^{-2x+10}$ .

Since both sides now have the same base, set the two exponents equal to one another and solve:

18x-30=-2x+10

Add 30 to both sides: 18x=-2x+40

Add to both sides: 20x=40

Divide both sides by 20: x=2

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Example Question #7 : Solving Exponential Equations

Solve $9^{4x+2}=27^{3x-4}$ .

Possible Answers:

$x=\frac{-16}{9}$

x=6

x=3

x=16

No solution

Correct answer:

x=16

Explanation:

Both 27 and 9 are powers of 3, therefore the equation can be rewritten as

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

Using the Distributive Property,

$3^{8x+4}=3^{9x-12}$ .

Now that both sides have the same base, set the two exponenents equal and solve.

8x+4=9x-12

Add 12 to both sides: 8x+16=9x

Subtract from both sides: 16=x

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Example Question #1 : Solving Exponential Equations

$(3)2^{x}=24$

Possible Answers:

x=1

x=0

x=4

x=-2

x=3

Correct answer:

x=3

Explanation:

The first step in thist problem is divide both sides by three: $2^{x}=8$ . Then, recognize that 8 could be rewritten with a base of 2 as well ( 2^3 ). Therefore, your answer is 3.

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Example Question #9 : Solving Exponential Equations

Solve for .

2^x=64

Possible Answers:

Correct answer:

Explanation:

Let's convert to base .

We know the following:

64=2*2*2*2*2*2

Simplify.

2*2*2*2*2*2=2^6

Solve.

x=6

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Example Question #10 : Solving Exponential Equations

Solve for .

3^x=2187

Possible Answers:

Correct answer:

Explanation:

Let's convert 2187 to base .

We know the following:

2187=3*3*3*3*3*3*3

Simplify.

3*3*3*3*3*3*3=3^7

Solve.

x=7 .

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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 #1 : Solving Exponential Equations

Example Question #1 : Solving Exponential Equations

Example Question #1 : Solving Exponential Equations

Example Question #1 : Solving Exponential Functions

Example Question #2 : Solving Exponential Functions

Example Question #11 : Solving Functions

Example Question #7 : Solving Exponential Equations

Example Question #1 : Solving Exponential Equations

Example Question #9 : Solving Exponential Equations

Example Question #10 : Solving Exponential Equations

All Algebra II Resources

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