Exponential Functions and Equations Practice Questions — SAT

If $5^{x+2} - 5^x = 120\sqrt{5}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 1.5

Answer

1.5 (or 3/2)

By factoring out the common term

5^x

from the left side of the equation, we get

5^x(5^2 - 1) = 120\sqrt{5}

, which simplifies to

24 \cdot 5^x = 120\sqrt{5}

. Dividing both sides by 24 isolates the exponential term:

5^x = 5\sqrt{5}

. Since

5\sqrt{5}

can be written as

5^1 \cdot 5^{0.5} = 5^{1.5}

, we have

5^x = 5^{1.5}

. Equating the exponents gives

x = 1.5

(or

3/2

).

Step-by-Step Solution

1

Factor out

5^x

from the left side of the equation.

5^x(5^2 - 1) = 120\sqrt{5}

To apply exponent rules to rewrite

5^{x+2}

as

5^x \cdot 5^2

and then factor out the common term

5^x

.

2

Simplify the constant term inside the parentheses.

24 \cdot 5^x = 120\sqrt{5}

Evaluating

5^2 - 1 = 25 - 1 = 24

simplifies the coefficient of the exponential expression.

3

Divide both sides of the equation by 24.

5^x = 5\sqrt{5}

To isolate the exponential term

5^x

on one side of the equation.

4

Express the right side as a single power of 5.

5^x = 5^{1.5}

Using exponent rules where

5\sqrt{5} = 5^1 \cdot 5^{0.5} = 5^{1.5}

so that both sides have the same base.

5

Equate the exponents of the common base 5.

x = 1.5

Since the bases on both sides of the equation are equal and positive, their exponents must be equal.

Key Concept

Solving exponential equations by factoring and rewriting terms using a common base.

Estimated Time:1m 30s

Question 2Question →

If $27^x = 9^{x + 1}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2

Answer

The value of

x

is

2

.

By rewriting

27

as

3^3

and

9

as

3^2

, the equation becomes

(3^3)^x = (3^2)^{x+1}

. Applying the power rule of exponents, this simplifies to

3^{3x} = 3^{2x+2}

. Since the bases are now the same, the exponents can be set equal to each other:

3x = 2x + 2

. Solving for

x

gives

x = 2

.

Step-by-Step Solution

1

Rewrite

27

and

9

as powers of

3

.

(3^3)^x = (3^2)^{x + 1}

To solve exponential equations with different bases, it is helpful to express them using a common base.

2

Apply the power of a power rule

(a^m)^n = a^{mn}

to simplify the exponents.

3^{3x} = 3^{2x + 2}

This simplifies the exponential expressions on both sides of the equation.

3

Set the exponents equal to each other.

3x = 2x + 2

Since the bases are equal (

3 = 3

), the exponents must also be equal for the equation to hold.

4

Solve the linear equation for

x

.

x = 2

Subtracting

2x

from both sides isolates the variable

x

.

Key Concept

Solving exponential equations by expressing both sides with a common base and equating exponents.

Estimated Time:45s

Question 3Question →

If $4^{x+2} \cdot 8^{x-1} = 16^{x+1}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

3

To solve the equation

4^{x+2} \cdot 8^{x-1} = 16^{x+1}

, rewrite all bases in terms of base 2:

(2^2)^{x+2} \cdot (2^3)^{x-1} = (2^4)^{x+1}

. Simplifying using the power rule yields

2^{2x+4} \cdot 2^{3x-3} = 2^{4x+4}

. Applying the product rule on the left side gives

2^{(2x+4)+(3x-3)} = 2^{5x+1}

. Equating the exponents gives

5x + 1 = 4x + 4

. Solving for

x

results in

x = 3

.

Step-by-Step Solution

1

Express the bases 4, 8, and 16 as powers of 2.

4^{x+2} = (2^2)^{x+2} = 2^{2x+4}

,

8^{x-1} = (2^3)^{x-1} = 2^{3x-3}

, and

16^{x+1} = (2^4)^{x+1} = 2^{4x+4}

Writing all parts of the equation with a common base allows the exponents to be compared directly.

2

Combine the terms on the left side by adding their exponents.

2^{2x+4} \cdot 2^{3x-3} = 2^{(2x+4) + (3x-3)} = 2^{5x+1}

According to the product rule of exponents,

b^m \cdot b^n = b^{m+n}

when the bases are the same.

3

Equate the exponents from both sides of the equation.

5x + 1 = 4x + 4

If two exponential expressions with the same positive base (other than 1) are equal, their exponents must be equal.

4

Solve the linear equation for

x

.

x = 3

Subtract

4x

and

1

from both sides to isolate the variable

x

.

Key Concept

Solving exponential equations by finding a common base and applying exponent laws.

Question 4Question →

A sample of a radioactive isotope decays such that its mass, in grams, is modeled by the function $M(t) = A \cdot d^{\frac{t}{12}}$ , where $t$ is the time in hours since the measurement began, and $A$ and $d$ are positive constants. The mass of the sample decreases by $75\%$ every $24$ hours. If the mass of the sample is $15$ grams when $t = 36$ , what was the initial mass, in grams, of the sample?

Show answer & explanation

Answer: 120

Answer

The initial mass of the sample was 120 grams.

The correct answer is 120. A decrease of

75\%

over 24 hours means the mass at

t+24

is

0.25

times the mass at

t

. According to the function,

M(t+24) = A \cdot d^{(t+24)/12} = A \cdot d^{t/12} \cdot d^2 = M(t) \cdot d^2

. Equating the two yields

d^2 = 0.25

, so

d = 0.5

. Substituting

t = 36

and

M(36) = 15

gives

15 = A \cdot (0.5)^{36/12} = A \cdot (0.5)^3 = 0.125A

. Solving for

A

gives

A = 120

.

Step-by-Step Solution

1

Relate the 24-hour decay rate to the exponent in the function to set up an equation for

d

.

d^2 = 0.25

Every 24 hours (

t

increases by 24), the mass decreases by

75\%

, so it becomes

25\%

(

0.25

) of its previous value. The exponent increases by

24/12 = 2

, multiplying the mass by

d^2

.

2

Solve for the decay base

d

.

d = 0.5

Since

d

is a positive constant, taking the square root of

0.25

gives

0.5

.

3

Substitute the given mass at

t = 36

into the model to solve for the initial mass

A

.

A = 120

Plugging

t = 36

and

M(36) = 15

into

M(t) = A \cdot (0.5)^{t/12}

gives

15 = A \cdot (0.5)^3

, which simplifies to

15 = 0.125A

.

Key Concept

Interpreting and solving exponential decay functions with fractional exponents

Alternative Method

Instead of solving for

d

first, recognize that

36

hours is exactly

1.5

intervals of

24

hours. Since the mass is multiplied by

0.25

(or

\frac{1}{4}

) every

24

hours, after

36

hours it will be multiplied by

(\frac{1}{4})^{1.5} = (\frac{1}{4})^{3/2} = \frac{1}{8}

of its initial value. Therefore,

15 = A \cdot \frac{1}{8} \implies A = 120

.

Estimated Time:3m 0s

Question 5Question →

The amount of a radioactive isotope remaining after $t$ days is modeled by the function $A(t) = A_0(0.64)^{\frac{t}{2}}$ , where $A_0$ is the initial amount of the isotope. If the function is rewritten in the form $A(t) = A_0(1 - r)^t$ , where $r$ is the daily decay rate, what is the value of $r$ ?

0.18

0.20

0.36

0.59

Show answer & explanation

Answer: 0.20

Answer

The daily decay rate

r

is

0.20

.

The correct value of

r

is

0.20

. By rewriting the given expression using exponent rules, we obtain

(0.64)^{t/2} = ((0.64)^{1/2})^t = (0.8)^t

. Setting the base equal to the target decay rate form gives

1 - r = 0.8

, which simplifies to

r = 0.20

.

Step-by-Step Solution

1

Apply the power of a power exponent rule,

(x^a)^b = x^{ab}

, to rewrite the exponential term.

A(t) = A_0\left(0.64^{1/2}\right)^t

This isolates the variable

t

as the exponent, allowing us to find the daily decay factor.

2

Evaluate the base

0.64^{1/2}

.

\sqrt{0.64} = 0.8

, so the expression becomes

A(t) = A_0(0.8)^t

.

Calculating the square root of

0.64

gives the daily decay factor of

0.8

.

3

Equate the daily decay factor

0.8

to the target form

1 - r

and solve for

r

.

1 - r = 0.8 \implies r = 0.20

Solving the equation gives the value of the daily decay rate.

Key Concept

Rewriting exponential equations by manipulating bases and exponents

Alternative Method

We can solve for

r

by choosing a specific value for

t

, such as

t = 2

. After 2 days, the remaining amount is

A(2) = A_0(0.64)^1 = 0.64A_0

. Using the target equation form, the remaining amount after 2 days is

A_0(1 - r)^2

. Setting the two expressions equal to each other gives

A_0(1 - r)^2 = 0.64A_0

. Dividing both sides by

A_0

gives

(1 - r)^2 = 0.64

. Taking the square root of both sides gives

1 - r = 0.8

, which yields

r = 0.20

.

Estimated Time:1m 30s

Question 6Question →

If $4^{2n-3} = \left(\frac{1}{8}\right)^{n-5}$ , what is the value of $n$ ?

Show answer & explanation

Answer: 3

Answer

3

Rewriting the bases 4 and

\frac{1}{8}

as powers of 2 gives

(2^2)^{2n-3} = (2^{-3})^{n-5}

. Applying the exponent power rule simplifies this to

2^{4n-6} = 2^{-3n+15}

. Since the bases are equal, the exponents must be equal:

4n-6 = -3n+15

. Solving this linear equation by adding

3n

and 6 to both sides gives

7n = 21

, which yields

n = 3

.

Step-by-Step Solution

1

Express the bases 4 and

\frac{1}{8}

as powers of the prime base 2.

4 = 2^2

and

\frac{1}{8} = 2^{-3}

Rewriting the bases with a common base of 2 allows for the application of exponent rules to solve the equation.

2

Substitute these bases back into the original equation.

(2^2)^{2n-3} = (2^{-3})^{n-5}

This sets up the equation to simplify the exponent terms.

3

Apply the power of a power rule,

(a^b)^c = a^{bc}

, by multiplying the exponents on both sides.

2^{2(2n-3)} = 2^{-3(n-5)} \implies 2^{4n-6} = 2^{-3n+15}

Multiplying the exponents simplifies the expressions to a single base raised to a single exponent on each side.

4

Equate the exponents since the bases are now equal, and solve for

n

.

4n-6 = -3n+15 \implies 7n = 21 \implies n = 3

If

b^x = b^y

where

b > 0

and

b \neq 1

, then

x = y

.

Key Concept

Solving exponential equations by expressing bases in terms of a common base and applying exponent rules.

Question 7Question →

If $3^{x - 2} = 81$ , what is the value of $x$ ?

2

4

6

29

Show answer & explanation

Answer: 6

Answer

6

To solve the equation

3^{x - 2} = 81

, we first express the number 81 as a power of 3, which is

3^4

. This gives us

3^{x - 2} = 3^4

. Since the bases are equal, we can set their exponents equal to each other, resulting in the equation

x - 2 = 4

. Solving for

x

by adding 2 to both sides gives the correct value of 6.

Step-by-Step Solution

1

Express both sides of the equation with a common base of 3.

3^{x - 2} = 3^4

Since 81 is equal to

3 \times 3 \times 3 \times 3

, it can be written as

3^4

.

2

Set the exponents equal to each other because the bases are now the same.

x - 2 = 4

If

b^y = b^z

for a positive base

b \neq 1

, then

y = z

.

3

Solve the linear equation for

x

by adding 2 to both sides of the equation.

x = 6

Adding 2 to both sides isolates the variable

x

.

Key Concept

Solving exponential equations by expressing both sides with a common base and equating their exponents.

Question 8Question →

A population of bacteria doubles every 3 hours. If the initial population of the bacteria is 500, what is the population of the bacteria after 9 hours?

Show answer & explanation

Answer: 4000

Answer

The population of the bacteria after 9 hours is 4,000.

The final population is calculated using the formula

P(t) = P_0 \times 2^{t/d}

, where

P_0

is the initial population of 500,

d

is the doubling period of 3 hours, and

t

is the total time of 9 hours. Evaluating this gives

P(9) = 500 \times 2^{9/3} = 500 \times 2^3 = 500 \times 8 = 4000

.

Step-by-Step Solution

1

Identify the initial population (

P_0

), doubling time (

d

), and total time (

t

).

P_0 = 500

,

d = 3

, and

t = 9

.

These parameters are required to set up the exponential growth model.

2

Calculate the number of doubling periods.

The number of doubling periods is

\frac{9}{3} = 3

.

The population doubles once for every 3-hour interval.

3

Calculate the final population using the exponential growth formula.

500 \times 2^3 = 500 \times 8 = 4000

.

Applying the 3 doubling cycles to the initial population of 500 yields the final population.

Key Concept

Exponential Growth Model

Estimated Time:1m 0s

Question 9Question →

A bank account is opened with an initial deposit of $800. The account balance increases by$ 5\% $each year. If no other transactions are made, which of the following functions best models the account balance,$ B(t) $, in dollars, after$ t$ years?

B(t) = 800(1.05)^t

B(t) = 800(0.05)^t

B(t) = 800(1.5)^t

B(t) = 800 + 40t

Show answer & explanation

Answer: $B(t) = 800(1.05)^t$

Answer

The function

B(t) = 800(1.05)^t

best models the account balance.

The initial deposit of

800 represents the initial value of the exponential function when

t = 0

. Since the account balance increases by

5\%

each year, the value is multiplied by a growth factor of

1 + 0.05 = 1.05

each year. Thus, the exponential function that models the balance after

t

years is

B(t) = 800(1.05)^t$.

Step-by-Step Solution

1

Identify the initial value of the exponential growth function.

The initial deposit is

800, so the initial value at

t = 0

is

800$.

An exponential model is written in the form

B(t) = a(b)^t

, where

a

represents the initial value.

2

Determine the growth factor based on the annual percentage increase.

The growth rate is

r = 5\% = 0.05

. The growth factor

b

is

1 + r = 1 + 0.05 = 1.05

.

To find the growth factor for an increasing quantity, add the growth rate as a decimal to 1.

3

Write the final exponential function by substituting the initial value and growth factor.

The function is

B(t) = 800(1.05)^t

.

Substitute

a = 800

and

b = 1.05

into the standard exponential form

B(t) = a(b)^t

.

Key Concept

Writing and interpreting exponential growth functions from a real-world context.

Question 10Question →

A scientist studying a sample of a radioactive isotope determines that its mass decays exponentially. The mass of the isotope, in grams, is modeled by the function $M(t) = a \cdot b^t$ , where $t$ is the time, in days, since the study began, and $a$ and $b$ are constants. The table below shows the mass of the isotope for selected values of $t$ .

$t$ (days)	$M(t)$ (grams)
0	128
1	96
2	72

What is the value of $b$ ?

Show answer & explanation

Answer: 0.75

Answer

0.75

To find the constant

b

in the exponential decay function

M(t) = a \cdot b^t

, we can use the given table values. For

t = 0

,

M(0) = a \cdot b^0 = a = 128

. For

t = 1

,

M(1) = a \cdot b^1 = a \cdot b = 96

. Substituting

a = 128

into the second equation gives

128b = 96

. Dividing both sides by

128

yields

b = \frac{96}{128} = 0.75

(or

\frac{3}{4}

). We can verify this with

t = 2

:

M(2) = 128 \cdot (0.75)^2 = 128 \cdot 0.5625 = 72

, which matches the table.

Step-by-Step Solution

1

Determine the initial value

a

using the data point for

t = 0

.

a = 128

Substituting

t = 0

into

M(t) = a \cdot b^t

gives

M(0) = a \cdot b^0 = a

. Since the table shows

M(0) = 128

,

a

must equal

128

.

2

Use the data point for

t = 1

to write an equation for

b

.

128b = 96

Substituting

t = 1

and

a = 128

into the function gives

M(1) = 128 \cdot b^1 = 128b

. The table shows

M(1) = 96

.

3

Solve the equation to find

b

.

b = 0.75

Dividing both sides of

128b = 96

by

128

yields

b = \frac{96}{128}

, which simplifies to

0.75

(or

\frac{3}{4}

).

Key Concept

Finding the decay factor (base) of an exponential function from a table of values.

Question 11Question →

If $9^{2x + 1} = 27^{x - 3}$ , what is the value of $x$ ?

-5

-4

-11

10

Show answer & explanation

Answer: $-11$

Answer

-11

To solve

9^{2x + 1} = 27^{x - 3}

, express 9 and 27 with a common base of 3:

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

. Apply the power of a power rule to get

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

, which simplifies to

3^{4x + 2} = 3^{3x - 9}

. Equating the exponents gives

4x + 2 = 3x - 9

. Solving for

x

by subtracting

3x

and

2

from both sides yields

x = -11

. Therefore, the value of

x

is

-11

.

Step-by-Step Solution

1

Rewrite both sides of the equation with a common base.

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

Since 9 and 27 are both powers of 3 (

9 = 3^2

and

27 = 3^3

), expressing them with the same base allows the application of exponent rules.

2

Apply the power of a power rule

(a^m)^n = a^{mn}

to simplify the exponents.

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

, which simplifies to

3^{4x+2} = 3^{3x-9}

Multiplying the inner exponent by the outer exponent simplifies the expression on both sides.

3

Set the exponents equal to each other.

4x + 2 = 3x - 9

Since the bases are equal and positive, the exponential expressions are equal if and only if their exponents are equal.

4

Solve the linear equation for

x

.

x = -11

Subtracting

3x

and

2

from both sides isolates the variable

x

.

Key Concept

Solving exponential equations by converting to a common base and equating exponents.

Question 12Question →

A wildlife biologist models the population of a certain bird species in a nature reserve using the function $P(t) = 120(1.05)^t$ , where $P(t)$ represents the number of birds $t$ years after the study began. What does the number $120$ represent in this model?

The initial number of birds in the reserve when the study began

The number of birds added to the reserve each year

The percent by which the bird population increases each year

The number of years it takes for the bird population to double

Show answer & explanation

Answer: The initial number of birds in the reserve when the study began

Answer

The initial number of birds in the reserve when the study began

In the exponential function model

P(t) = a \cdot b^t

, the coefficient

a

is the value of the function at

t = 0

. Evaluating

P(0)

yields

120(1.05)^0 = 120

, which corresponds to the initial number of birds in the reserve when the study began.

Step-by-Step Solution

1

Identify the standard form of the exponential growth function

P(t) = a \cdot b^t

, where

a

represents the initial value and

b

represents the growth factor.

This helps map the given constants to their mathematical meanings.

2

Substitute

t = 0

to find the starting population

P(0) = 120(1.05)^0 = 120 \cdot 1 = 120

.

Finding the value at

t = 0

defines the initial state of the model.

3

Interpret the initial value in the context of the problem

The number

120

represents the number of birds present at the start of the study.

This links the mathematical result to the real-world scenario.

Key Concept

Interpreting components of an exponential function in context

Question 13Question →

If $8^{x+2} = \left(\frac{1}{4}\right)^{1-3x}$ , what is the value of $x$ ?

\frac{8}{3}

\frac{4}{3}

-\frac{4}{9}

5

Show answer & explanation

Answer: $\frac{8}{3}$

Answer

\frac{8}{3}

To solve the equation

8^{x+2} = \left(\frac{1}{4}\right)^{1-3x}

, both bases can be written as powers of

2

because

8 = 2^3

and

\frac{1}{4} = 2^{-2}

. Substituting these values into the equation yields

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

. Applying the power rule of exponents,

(a^m)^n = a^{mn}

, we multiply the exponents to get

2^{3x+6} = 2^{-2+6x}

. Since the bases are equal, we can set the exponents equal to each other:

3x + 6 = -2 + 6x

. Solving this linear equation by subtracting

3x

from both sides and adding

2

to both sides gives

8 = 3x

, which simplifies to

x = \frac{8}{3}

. Thus, the option with the value

\frac{8}{3}

is correct.

Step-by-Step Solution

1

Express both bases in the equation,

8

and

\frac{1}{4}

, as powers of

2

.

The base

8

is written as

2^3

and the base

\frac{1}{4}

is written as

2^{-2}

, yielding the equation

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

.

Expressing exponential terms with a common base is necessary to equate and solve their exponents.

2

Apply the power of a power exponent rule,

(a^m)^n = a^{mn}

, to simplify both sides of the equation.

The equation becomes

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

, which simplifies to

2^{3x+6} = 2^{-2+6x}

.

This simplifies each side to a single base with a single exponent.

3

Since the bases are equal, set the exponents equal to each other and solve the resulting linear equation for

x

.

3x + 6 = -2 + 6x \implies 8 = 3x \implies x = \frac{8}{3}

.

Two exponential expressions with the same positive base are equal if and only if their exponents are equal.

Key Concept

Solving exponential equations by expressing terms with a common base and applying exponent rules.

Question 14Question →

If $25^a = 125^b$ , where $a$ and $b$ are positive constants, what is the value of $\frac{a}{b}$ ?

Show answer & explanation

Answer: 1.5

Answer

1.5

To find the value of

\frac{a}{b}

, we can rewrite the bases of the equation

25^a = 125^b

using a common base of 5:

(5^2)^a = (5^3)^b

. Applying the exponent power rule gives

5^{2a} = 5^{3b}

. Since the bases are identical, their exponents must be equal, so

2a = 3b

. Dividing both sides of this equation by

2b

yields the ratio

\frac{a}{b} = \frac{3}{2}

, which is

1.5

.

Step-by-Step Solution

1

Express the bases as powers of 5

(5^2)^a = (5^3)^b

To solve the equation, express both sides with a common base of 5 since

25 = 5^2

and

125 = 5^3

.

2

Apply the power rule of exponents

5^{2a} = 5^{3b}

According to the exponent rules,

(x^m)^n = x^{mn}

.

3

Set the exponents equal to each other

2a = 3b

Because the bases on both sides of the equation are equal and positive, their exponents must also be equal.

4

Solve for the ratio

\frac{a}{b} = 1.5

Divide both sides of the equation

2a = 3b

by

2b

to isolate the ratio

\frac{a}{b}

.

Key Concept

Solving exponential equations by expressing bases in terms of a common base and equating exponents.

Question 15Question →

The function $f$ is defined by $f(x) = a \cdot 3^x$ , where $a$ is a constant. If $f(2) = 45$ , what is the value of $f(1)$ ?

Show answer & explanation

Answer: 15

Answer

15

To find the value of

f(1)

, we first determine the value of the constant

a

. We are given that

f(2) = 45

, so substituting

x = 2

into the function definition

f(x) = a \cdot 3^x

gives

45 = a \cdot 3^2

. Simplifying

3^2

to

9

yields

45 = 9a

, which means

a = 5

. Now that we know

a = 5

, we can write the function as

f(x) = 5 \cdot 3^x

. To find

f(1)

, we substitute

x = 1

into this equation, yielding

f(1) = 5 \cdot 3^1 = 15

.

Step-by-Step Solution

1

Substitute the given point

(2, 45)

into the function equation to solve for

a

.

a = 5

Since

f(2) = 45

, we have

45 = a \cdot 3^2 = 9a

, which gives

a = 5

.

2

Evaluate the function at

x = 1

using the value of

a = 5

.

f(1) = 15

Substituting

a = 5

and

x = 1

into

f(x) = a \cdot 3^x

gives

f(1) = 5 \cdot 3^1 = 15

.

Key Concept

Evaluating and solving exponential functions given initial conditions or points.

Question 16Question →

If $3^{x+1} = k$ , where $k > 0$ , which of the following is equivalent to $27^{x-1}$ ?

\frac{k^3}{729}

\frac{k^3}{81}

\frac{k^3}{27}

\frac{k^3}{9}

Show answer & explanation

Answer: $\frac{k^3}{729}$

Answer

The expression \frac{k^3}{729}

To express

27^{x-1}

in terms of

k

, we can rewrite both the given equation and the target expression using base

3

. The given equation

3^{x+1} = k

can be adjusted to find the value of

3^{x-1}

by dividing both sides by

3^2

, which yields

3^{x-1} = \frac{k}{9}

. The target expression

27^{x-1}

can be rewritten as

(3^3)^{x-1} = (3^{x-1})^3

. Substituting

3^{x-1} = \frac{k}{9}

into this expression gives

\left(\frac{k}{9}\right)^3 = \frac{k^3}{729}

.

Step-by-Step Solution

1

Express the given equation in terms of a simpler base

3

exponent.

3^{x-1} = \frac{k}{9}

We start with the given equation

3^{x+1} = k

. To relate this to the exponent

x-1

, we divide both sides of the equation by

3^2 = 9

:

3^{x-1} = \frac{3^{x+1}}{3^2} = \frac{k}{9}

.

2

Rewrite the target expression

27^{x-1}

with base

3

.

27^{x-1} = (3^{x-1})^3

Since

27 = 3^3

, we can rewrite the target expression as

27^{x-1} = (3^3)^{x-1} = 3^{3(x-1)} = (3^{x-1})^3

.

3

Substitute the expression for

3^{x-1}

from Step 1 into the expression from Step 2.

\frac{k^3}{729}

Substituting

3^{x-1} = \frac{k}{9}

into

(3^{x-1})^3

gives

\left(\frac{k}{9}\right)^3 = \frac{k^3}{9^3} = \frac{k^3}{729}

.

Key Concept

Manipulating exponential equations by expressing terms with a common base and applying exponent rules.

Question 17Question →

If $2^{2x+2} - 2^{2x} = 12$ for some real number $x$ , what is the value of $2^{4x}$ ?

Show answer & explanation

Answer: 16

Answer

16

By applying the product rule of exponents, the expression

2^{2x+2}

can be rewritten as

2^{2x} \cdot 2^2

, or

4 \cdot 2^{2x}

. Substituting this into the given equation yields

4 \cdot 2^{2x} - 2^{2x} = 12

. Factoring out

2^{2x}

gives

2^{2x}(4 - 1) = 12

, which simplifies to

3 \cdot 2^{2x} = 12

. Dividing both sides of the equation by 3 results in

2^{2x} = 4

. Since

2^{4x}

can be written as

(2^{2x})^2

, substituting 4 for

2^{2x}

gives

4^2 = 16

. Alternatively, solving

2^{2x} = 4

gives

2x = 2

, which means

x = 1

. Substituting

x = 1

into

2^{4x}

yields

2^{4(1)} = 2^4 = 16

.

Step-by-Step Solution

1

Rewrite

2^{2x+2}

using exponent properties.

2^{2x+2} = 2^{2x} \cdot 2^2 = 4 \cdot 2^{2x}

To express both exponential terms with the same base power, allowing them to be factored or combined.

2

Substitute this back into the equation and factor out the common term

2^{2x}

.

2^{2x}(4 - 1) = 12 \implies 3 \cdot 2^{2x} = 12

To isolate the exponential expression.

3

Solve for

2^{2x}

by dividing both sides of the equation by 3.

2^{2x} = 4

To find the value of the exponential term.

4

Express

2^{4x}

in terms of

2^{2x}

and evaluate.

2^{4x} = (2^{2x})^2 = 4^2 = 16

To find the final requested value using the exponent rule

(a^m)^n = a^{mn}

.

Key Concept

Factoring exponential equations and applying power of a power exponent rules

Question 18Question →

If $3^{x+4} = 9^x$ , what is the value of $x$ ?

4

2

-4

\frac{4}{3}

Show answer & explanation

Answer: 4

Answer

4

To solve the equation

3^{x+4} = 9^x

, we rewrite the base 9 as

3^2

, which gives

3^{x+4} = (3^2)^x

. Applying the exponent rule

(a^m)^n = a^{mn}

, we get

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

. Since the bases are now the same, we can set the exponents equal to each other:

x+4 = 2x

. Subtracting

x

from both sides yields the solution

4

.

Step-by-Step Solution

1

Express both sides of the equation with a common base.

Since

9 = 3^2

, the equation

3^{x+4} = 9^x

can be rewritten as

3^{x+4} = (3^2)^x

. Using the power of a power rule, this becomes

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

.

To solve an exponential equation, it is helpful to have the same base on both sides so that the exponents can be equated.

2

Set the exponents equal to each other.

x + 4 = 2x

Since the bases are equal and positive (and not equal to 1), their exponents must be equal.

3

Solve the linear equation for

x

.

4 = x

(or

x = 4

)

Subtract

x

from both sides of the equation to isolate the variable.

Key Concept

Solving exponential equations by expressing terms with a common base and equating their exponents.

Estimated Time:45s

Question 19Question →

If $4^{x - 1} = 8$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2.5

Answer

2.5 (or 5/2)

The correct answer is 2.5 (or 5/2). By expressing 4 as

2^2

and 8 as

2^3

, the equation is rewritten as

(2^2)^{x-1} = 2^3

. Applying the exponent power rule simplifies this to

2^{2x-2} = 2^3

. Equating the exponents yields the linear equation

2x-2 = 3

. Solving for

x

gives

2x = 5

, which results in

x = 2.5

or

5/2

.

Step-by-Step Solution

1

Rewrite both sides of the equation with a common base of 2.

(2^2)^{x-1} = 2^3

Expressing both bases as powers of 2 allows us to equate the exponents later.

2

Apply the power of a power exponent rule

(a^m)^n = a^{mn}

to the left side.

2^{2x-2} = 2^3

Multiplying the exponent 2 by the exponent

(x-1)

simplifies the expression to

2(x-1) = 2x-2

.

3

Equate the exponents and solve the linear equation for

x

.

x = 2.5

Since the bases are identical, their exponents must be equal, giving

2x-2 = 3

, which simplifies to

2x = 5

.

Key Concept

Solving exponential equations by finding a common base.

Question 20Question →

If $3^{2x - 1} = 27$ , what is the value of $x$ ?

1

5

2

14

Show answer & explanation

Answer: 2

Answer

2

To solve the equation

3^{2x - 1} = 27

, we first express 27 as a power of 3, which is

3^3

. Since the bases are the same, we set the exponents equal to each other to get

2x - 1 = 3

. Adding 1 to both sides gives

2x = 4

, and dividing both sides by 2 gives the value of

x

as 2.

Step-by-Step Solution

1

Express 27 with a base of 3

3^{2x - 1} = 3^3

To solve an exponential equation, we need to write both sides of the equation with a common base.

2

Set the exponents equal to each other

2x - 1 = 3

Since the bases are equal, their exponents must also be equal.

3

Solve the linear equation for

x

x = 2

Add 1 to both sides to get

2x = 4

, and then divide both sides by 2 to isolate

x

.

Key Concept

Solving exponential equations by finding a common base

Estimated Time:45s

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