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SAT II Math I : Solving Exponential Functions

Study concepts, example questions & explanations for SAT II Math I

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All SAT II Math I Resources

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Example Questions

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Example Question #1 : Solving Exponential Equations

Solve for :

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

Possible Answers:

x=2

x=5

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

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

No solution

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 #8 : Solving Exponential Equations

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

Possible Answers:

x=0

x=1

x=3

x=4

x=-2

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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Example Question #11 : Solving Exponential Equations

Solve for .

2^x*2^x=2^6

Possible Answers:

Correct answer:

Explanation:

When multiplying exponents with the same base, we will apply the power rule of exponents:

(a^m)^n=a^m^n

We will simply add the exponents and keep the base the same.

2x=6

x=3

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Example Question #11 : Solving Exponential Equations

Solve for .

$3^7*3^{x+3}=3^{24}$

Possible Answers:

Correct answer:

Explanation:

When multiplying exponents with the same base, we will apply the power rule of exponents:

(a^m)^n=a^m^n

We will simply add the exponents and keep the base the same.

7+x+3=24

Simplify.

10+x=24

Solve.

x=14

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Example Question #13 : Solving Exponential Equations

Solve for .

2^x+2^x=2^7

Possible Answers:

3.5

2.5

Correct answer:

Explanation:

When adding exponents with the same base, we need to see if we can factor out the numbers of the base.

In this case, let's factor out 2^x .

We get the following:

2^x(1+1)=2^x(2)

Since we are now multiplying with the same base, we get the following expression:

$2^{x+1}$

Now we have the same base and we just focus on the exponents.

The equation is now:

x+1=7

Solve.

x=6

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Example Question #14 : Solving Exponential Equations

Solve for .

3^x+3^x+3^x=27^2

Possible Answers:

$\frac{2}{3}$

$\frac{5}{3}$

Correct answer:

Explanation:

First, we need to convert 27^2 to base .

We know 27=3^3 .

Therefore we can write the following expression:

(3^3)^2=3^6 .

Next, when we add exponents of the same base, we need to see if we can factor out terms.

In this case, let's factor out 3^x .

We get the following:

3^x(1+1+1)=3^x(3) .

Since we are now multiplying with the same base, we get the following expression:

$3^{x+1}$ .

Now we have the same base and we just focus on the exponents.

The equation is now:

x+1=6

Solve.

x=5

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Example Question #1 : Graphing Exponential Functions

Give the -intercept of the graph of the equation $f(x) = 2 ^{x- 7} + 4$ .

Possible Answers:

(3.9921875, 0)

(5,0)

(4.0078125, 0)

(9,0)

The graph has no -intercept.

Correct answer:

The graph has no -intercept.

Explanation:

Set f(x) = 0 and solve for

$f(x) = 2 ^{x- 7} + 4$

$2 ^{x- 7} + 4 = 0$

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

We need not work further. It is impossible to raise a positive number 2 to any real power to obtain a negative number. Therefore, the equation has no solution, and the graph of has no -intercept.

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SAT II Math I : Solving Exponential Functions

Study concepts, example questions & explanations for SAT II Math I

All SAT II Math I Resources

Example Questions

Example Question #1 : Solving Exponential Equations

Example Question #7 : Solving Exponential Equations

Example Question #8 : Solving Exponential Equations

Example Question #9 : Solving Exponential Equations

Example Question #10 : Solving Exponential Equations

Example Question #11 : Solving Exponential Equations

Example Question #11 : Solving Exponential Equations

Example Question #13 : Solving Exponential Equations

Example Question #14 : Solving Exponential Equations

Example Question #1 : Graphing Exponential Functions

All SAT II Math I Resources

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