ACT Math Logarithms and Exponential Functions Quiz

1. If 2^x = 32, what is the value of x?

4

5

6

8

To solve the equation $2^x = 32$, we can express 32 as a power of 2. Since $32 = 2^5$, we can equate the exponents: $x = 5$. Thus, the value of $x$ is 5, as it satisfies the original equation.

Explanation

To solve the equation $2^x = 32$, we can express 32 as a power of 2. Since $32 = 2^5$, we can equate the exponents: $x = 5$. Thus, the value of $x$ is 5, as it satisfies the original equation.

2. What is log₃(81) equal to?

3

4

9

27

Logarithms represent the exponent to which a base must be raised to produce a given number. In this case, log₃(81) asks for the exponent that 3 must be raised to in order to equal 81. Since 3^4 = 81, log₃(81) equals 4.

Explanation

Logarithms represent the exponent to which a base must be raised to produce a given number. In this case, log₃(81) asks for the exponent that 3 must be raised to in order to equal 81. Since 3^4 = 81, log₃(81) equals 4.

3. Solve: 3^(2x) = 27. What is x?

1

1.5

2

3

To solve the equation 3^(2x) = 27, recognize that 27 can be expressed as 3^3. This leads to the equation 3^(2x) = 3^3. Since the bases are equal, set the exponents equal: 2x = 3. Solving for x gives x = 3/2, which simplifies to 1.5.

Explanation

To solve the equation 3^(2x) = 27, recognize that 27 can be expressed as 3^3. This leads to the equation 3^(2x) = 3^3. Since the bases are equal, set the exponents equal: 2x = 3. Solving for x gives x = 3/2, which simplifies to 1.5.

4. If log_b(64) = 2, what is the base b?

4

6

8

16

To solve log_b(64) = 2, we rewrite it in exponential form: b^2 = 64. By calculating the square root of 64, we find b = 8. Thus, the base b that satisfies the equation is 8.

Explanation

To solve log_b(64) = 2, we rewrite it in exponential form: b^2 = 64. By calculating the square root of 64, we find b = 8. Thus, the base b that satisfies the equation is 8.

5. Which expression is equivalent to log(x) + log(y)?

Log(x + y)

Log(xy)

Log(x/y)

Log(x^y)

The expression log(x) + log(y) can be simplified using the logarithmic property that states the sum of logarithms is equal to the logarithm of the product. Therefore, log(x) + log(y) is equivalent to log(xy), which represents the logarithm of the product of x and y.

Explanation

The expression log(x) + log(y) can be simplified using the logarithmic property that states the sum of logarithms is equal to the logarithm of the product. Therefore, log(x) + log(y) is equivalent to log(xy), which represents the logarithm of the product of x and y.

6. A bacteria population grows according to P(t) = 100 · 2^t, where t is time in hours. How many bacteria are present after 3 hours?

200

400

800

1600

To find the bacteria population after 3 hours, substitute $ t = 3 $ into the equation $ P(t) = 100 \cdot 2^t $. This gives $ P(3) = 100 \cdot 2^3 = 100 \cdot 8 = 800 $. Thus, there are 800 bacteria present after 3 hours.

Explanation

To find the bacteria population after 3 hours, substitute $ t = 3 $ into the equation $ P(t) = 100 \cdot 2^t $. This gives $ P(3) = 100 \cdot 2^3 = 100 \cdot 8 = 800 $. Thus, there are 800 bacteria present after 3 hours.

7. Simplify: log(1000) if the base is 10.

1

2

3

10

To simplify log(1000) with base 10, we determine what power of 10 equals 1000. Since 10^3 = 1000, it follows that log(1000) = 3. Thus, the logarithm of 1000 to the base 10 is 3.

Explanation

To simplify log(1000) with base 10, we determine what power of 10 equals 1000. Since 10^3 = 1000, it follows that log(1000) = 3. Thus, the logarithm of 1000 to the base 10 is 3.

8. Solve for x: 5^x = 125.

2

3

4

5

To solve the equation $5^x = 125$, recognize that 125 can be expressed as a power of 5: $125 = 5^3$. Thus, we can rewrite the equation as $5^x = 5^3$. Since the bases are the same, we can equate the exponents, leading to $x = 3$.

Explanation

To solve the equation $5^x = 125$, recognize that 125 can be expressed as a power of 5: $125 = 5^3$. Thus, we can rewrite the equation as $5^x = 5^3$. Since the bases are the same, we can equate the exponents, leading to $x = 3$.

9. Which is equivalent to 2 log(x) - log(y)?

Log(x²/y)

Log(2x - y)

Log(x²y)

Log(x/2y)

Using logarithmic properties, specifically the power rule and quotient rule, we can simplify the expression. The term 2 log(x) can be rewritten as log(x²), and when combined with -log(y), it becomes log(x²) - log(y). This simplifies to log(x²/y), demonstrating the equivalence.

Explanation

Using logarithmic properties, specifically the power rule and quotient rule, we can simplify the expression. The term 2 log(x) can be rewritten as log(x²), and when combined with -log(y), it becomes log(x²) - log(y). This simplifies to log(x²/y), demonstrating the equivalence.

10. If an investment doubles every 5 years and starts at $1000, what is the formula?

A = 1000 · 2^(t/5)

A = 1000 · 2^5t

A = 2000 · t

A = 1000 + 2t

The formula A = 1000 · 2^(t/5) represents exponential growth, where A is the amount after time t in years. The investment starts at $1000 and doubles every 5 years, meaning the exponent t/5 indicates the number of 5-year periods that have passed, effectively doubling the initial amount each time.

Explanation

The formula A = 1000 · 2^(t/5) represents exponential growth, where A is the amount after time t in years. The investment starts at $1000 and doubles every 5 years, meaning the exponent t/5 indicates the number of 5-year periods that have passed, effectively doubling the initial amount each time.

11. Evaluate: log_2(8) + log_2(4).

3

4

5

7

To evaluate log_2(8) + log_2(4), first find the individual logarithms: log_2(8) equals 3 since 2^3 = 8, and log_2(4) equals 2 since 2^2 = 4. Adding these results gives 3 + 2 = 5. Thus, the final answer is 5.

Explanation

To evaluate log_2(8) + log_2(4), first find the individual logarithms: log_2(8) equals 3 since 2^3 = 8, and log_2(4) equals 2 since 2^2 = 4. Adding these results gives 3 + 2 = 5. Thus, the final answer is 5.

12. Solve: 4^x = 1/16. What is x?

-2

-1

1

2

To solve the equation $4^x = \frac{1}{16}$, we can express $16$ as $4^2$, so $\frac{1}{16} = 4^{-2}$. This gives us $4^x = 4^{-2}$. Since the bases are equal, we can set the exponents equal to each other: $x = -2$. Thus, the solution is $x = -2$.

Explanation

To solve the equation $4^x = \frac{1}{16}$, we can express $16$ as $4^2$, so $\frac{1}{16} = 4^{-2}$. This gives us $4^x = 4^{-2}$. Since the bases are equal, we can set the exponents equal to each other: $x = -2$. Thus, the solution is $x = -2$.