ACT Math Logarithms | Practice Problems with Step-by-Step Explanations

ACT Math Logarithms: 31 Practice Problems with Step-by-Step Explanations

Logarithms tend to intimidate students more than almost any other ACT Math topic, but nearly every question comes down to one core move: converting between logarithmic form and exponential form. The statement log_b x = y means exactly the same thing as b^y = x, and being able to flip fluidly between these two forms unlocks the majority of problems in this chapter, from the simplest "solve for x" question to multi-step equations involving natural logs (ln) and change of base.

Beyond that core conversion, three logarithm properties come up constantly: the product property (log(AB) = log A + log B), the quotient property (log(A/B) = log A − log B), and the power property (log(Aⁿ) = n·log A). These let you combine or split logarithmic expressions, which is especially useful on problems that give you log values as variables (like "log_a x = v") and ask you to build up a more complex expression from them. The natural log, ln(x), is simply log base e, and it follows all the same rules — the only difference is the base.

Common traps include forgetting that a logarithm's base must be positive and not equal to 1, and that the argument of a logarithm (the thing being logged) must be positive — which matters most on problems where x appears in the base or the argument, since some solutions to the algebra turn out to be extraneous once you check them against these restrictions. Another frequent trap is misapplying the power property in reverse, or mixing up log(A) − log(B) with log(A−B), which are not the same thing. Work through the 31 problems below, then click to reveal each step-by-step explanation.

On this page:

Problem 1 — Log to exponential form
Problem 2 — Log to exponential form, negative
Problem 3 — Evaluating a logarithm
Problem 4 — Solving for the base
Problem 5 — Log of a radical
Problem 6 — Combining log values
Problem 7 — Solving for the base
Problem 8 — Quotient property
Problem 9 — Solving a log equation
Problem 10 — Log of a radical
Problem 11 — Solving a natural log equation
Problem 12 — Solving a log equation
Problem 13 — Estimating a logarithm
Problem 14 — Log equation with a quadratic
Problem 15 — Log to exponential form, fractional
Problem 16 — Intersection of two functions
Problem 17 — Solving a natural log equation
Problem 18 — Log with a variable base
Problem 19 — Solving a natural log equation
Problem 20 — Integer constraints on a log equation
Problem 21 — Sign and parity from a log equation
Problem 22 — Log values from a table
Problem 23 — Sign and parity from a log equation
Problem 24 — Real-world log model
Problem 25 — Combining log properties
Problem 26 — Building an expression from log values
Problem 27 — Real-world exponential growth
Problem 28 — Range from integer constraints
Problem 29 — Building an expression from log values
Problem 30 — Solving a complex log equation
Problem 31 — Inverse of an exponential function

Practice Problems

Problem 1

When log₅ x = 2, what is x?

A) 2.5
B) 5
C) 15
D) 25

Correct Answer: D

Rewrite the logarithmic equation in exponential form: log₅ x = 2 means 5² = x.

So x = 25.

Problem 2

When log₂ x = −4, what is x?

A) −16
B) −8
C) 1/8
D) 1/16

Correct Answer: D

Rewrite in exponential form: log₂ x = −4 means 2⁻⁴ = x.

2⁻⁴ = 1/2⁴ = 1/16.

Problem 3

What is 4(log₄ 16) equal to?

A) 1
B) 2
C) 4
D) 8

Correct Answer: D

First evaluate log₄ 16. Since 4² = 16, log₄ 16 = 2.

Multiply by 4: 4 × 2 = 8.

Problem 4

When log_x (1/125) = −3, what is x?

A) 1/25
B) 1/5
C) 3
D) 5

Correct Answer: D

Rewrite in exponential form: x⁻³ = 1/125, which means x³ = 125.

Since 5³ = 125, x = 5.

Problem 5

What is the value of log₃ √27?

A) 1/2
B) 3/2
C) √2
D) 1

Correct Answer: B

Rewrite √27 as a power of 3: √27 = 27^1/2 = (3³)^1/2 = 3^3/2.

So log₃(3^3/2) = 3/2.

Problem 6

What is the value of 5log₂ 8 − 2log₃ 9?

A) 3
B) 5
C) 11
D) 19

Correct Answer: C

Evaluate each logarithm: log₂ 8 = 3 (since 2³=8), and log₃ 9 = 2 (since 3²=9).

Substitute: 5(3) − 2(2) = 15 − 4 = 11.

Problem 7

What is the value of the positive real number x such that log_x(1/9) = −2?

A) 3
B) 9
C) 1/3
D) 1/9

Correct Answer: A

Rewrite in exponential form: x⁻² = 1/9, which means x² = 9.

Since x must be positive, x = 3.

Problem 8

For positive real numbers A and B, log(B/A) = ?

A) log B / log A
B) log(B−A)
C) log B − log A
D) log A − log B

Correct Answer: C

This is the quotient property of logarithms: the log of a quotient equals the difference of the logs, with the numerator's log listed first.

log(B/A) = log B − log A.

Problem 9

When log₁₁ 121^3x = 2, what is x?

A) 1/11
B) 1/3
C) 1/2
D) 1

Correct Answer: B

Rewrite 121 as a power of 11: 121 = 11², so 121^3x = (11²)^3x = 11^6x.

The equation becomes log₁₁(11^6x) = 2, so 6x = 2, giving x = 1/3.

Problem 10

log₃ ³√9 = ?

A) 0
B) 2/3
C) 1
D) ³√3

Correct Answer: B

Rewrite ³√9 as a power of 3: ³√9 = 9^1/3 = (3²)^1/3 = 3^2/3.

So log₃(3^2/3) = 2/3.

Problem 11

If ln(x+4) = 2, then x = ?

A) √(e−4)
B) √(x²−4)
C) 2e−4
D) e²−4

Correct Answer: D

Since ln is log base e, rewrite in exponential form: x + 4 = e².

Solve for x: x = e² − 4.

Problem 12

What real value of x satisfies the equation log₃ 27⁴ = 6x?

A) 2
B) 3
C) 6
D) 27

Correct Answer: A

Rewrite 27 as a power of 3: 27 = 3³, so 27⁴ = (3³)⁴ = 3¹².

The equation becomes log₃(3¹²) = 6x, so 12 = 6x, giving x = 2.

Problem 13

The value of log₈ 8^10/3 is between which of the following pairs of integers?

A) 0 and 1
B) 2 and 3
C) 3 and 4
D) 5 and 6

Correct Answer: C

Since the base of the logarithm matches the base of the power exactly, log₈ 8^10/3 simplifies directly to 10/3.

10/3 ≈ 3.33, which falls between 3 and 4.

Problem 14

When log₈ 1 = x² − 6x + 9, what is x?

A) 0
B) 2
C) 3
D) 6

Correct Answer: C

Any log of 1 (in any valid base) equals 0, since base⁰ = 1. So log₈ 1 = 0.

Set the quadratic equal to 0: x² − 6x + 9 = 0, which factors as (x−3)² = 0, giving x = 3.

Problem 15

If log₈ x = −2/3, then x = ?

A) −4
B) −1/4
C) 1/12
D) 1/4

Correct Answer: D

Rewrite in exponential form: x = 8^−2/3. First evaluate 8^1/3 = 2 (the cube root of 8), then apply the outer exponent: 8^2/3 = 2² = 4.

Since the original exponent is negative, take the reciprocal: 8^−2/3 = 1/4.

Problem 16

f(x) = ln(0.5x−2) + 3, and g(x) = 6. If the graphs of the two functions above intersect at the point (a,b), what is the value of a?

A) 2e³ + 4
B) e³ − 2
C) ln(3) + 2
D) ln(1) − 3

Correct Answer: A

At the intersection, f(x) = g(x): ln(0.5x−2) + 3 = 6. Subtract 3 from both sides: ln(0.5x−2) = 3.

Rewrite in exponential form: 0.5x − 2 = e³. Add 2 to both sides: 0.5x = e³ + 2. Multiply both sides by 2: x = 2e³ + 4.

Problem 17

If ln(2x−3) = 2, then x = ?

A) ln(2) − 1
B) ln(5)
C) 2e² − 3
D) (e²+3)/2

Correct Answer: D

Rewrite in exponential form: 2x − 3 = e². Add 3 to both sides: 2x = e² + 3.

Divide by 2: x = (e²+3)/2.

Problem 18

If log_(x+2)(x²+8) = 2, then x = ?

A) −1
B) 0
C) 1
D) 2

Correct Answer: C

Rewrite in exponential form: (x+2)² = x²+8. Expand the left side: x²+4x+4 = x²+8.

Subtract x² from both sides: 4x+4 = 8, so 4x = 4, giving x = 1. Checking the base: x+2 = 3, which is positive and not equal to 1, so this is a valid solution.

Problem 19

If ln(x²+10) = 4, what is x?

A) e⁴ − 10
B) √(e⁴−10)
C) e² − √10
D) ln(e²−10)

Correct Answer: B

Rewrite in exponential form: x² + 10 = e⁴. Subtract 10 from both sides: x² = e⁴ − 10.

Take the square root of both sides: x = √(e⁴−10).

Problem 20

For positive integers a and b where a < 9, log_a 9 = b. What is the value of a/b?

A) 1
B) 2/3
C) 3/2
D) 2

Correct Answer: C

Rewrite in exponential form: a^b = 9. Since a and b must be positive integers with a < 9, test small values: a = 3, b = 2 works, since 3² = 9.

No other integer pair with a < 9 satisfies a^b = 9, so a/b = 3/2.

Problem 21

Let x be an even integer greater than 3 and let a = x − 2. The equation log_a b = x is true. Which of the following describes every value of b?

A) Positive and odd
B) Positive and even
C) Negative and odd
D) Negative and even

Correct Answer: B

Rewrite in exponential form: a^x = b. Since x is even, a = x−2 is also even (even minus even is even). So b is an even number raised to an even power.

An even base raised to any positive integer power is always positive and even, so b is positive and even.

Problem 22

The value of log(x), to the nearest 0.1, is given in the table below for 4 different values of x. To the nearest 0.1, what is the value of log(5 × 10⁴⁰⁰)?

x	log(x)
5	0.7
50	1.7
500	2.7
5,000	3.7

A) 0.7
B) 280
C) 400.7
D) 403.7

Correct Answer: C

The table reveals a pattern: each time x is multiplied by 10, log(x) increases by exactly 1 (0.7, then 1.7, then 2.7, then 3.7). This makes sense since log(5×10ⁿ) = log(5) + n ≈ 0.7 + n.

Applying the same pattern to 5 × 10⁴⁰⁰: log(5 × 10⁴⁰⁰) ≈ 0.7 + 400 = 400.7.

Problem 23

If x is a positive even integer, a = x + 3, and log_a b = x, which of the following describes every value of b?

A) Positive and even
B) Positive and odd
C) Negative and even
D) Negative and odd

Correct Answer: B

Rewrite in exponential form: a^x = b. Since x is even, a = x+3 is odd (even plus odd is odd). So b is an odd number raised to an even power.

An odd base raised to any positive integer power is always positive and odd, so b is positive and odd.

Problem 24

The noise rating, r, of a water pump is modeled by the equation r = 20log(P/K), where P is the water pressure and K is a constant. What is the noise rating of a water pump whose pressure is 100 times the value of K?

A) 2
B) 20
C) 40
D) 400

Correct Answer: C

Substitute P = 100K: r = 20log(100K/K) = 20log(100).

Since log(100) = 2 (as 10²=100): r = 20(2) = 40.

Problem 25

Which of the following is equivalent to the expression log₇ 90 + log₇ 96 − 3log₇ 6?

A) log₇ 40
B) log₇ 168
C) log₇ 1440
D) log₇ 8424

Correct Answer: A

Use the product property to combine the addition: log₇ 90 + log₇ 96 = log₇(90×96) = log₇ 8640. Use the power property on the last term: 3log₇ 6 = log₇(6³) = log₇ 216.

Use the quotient property to combine everything: log₇ 8640 − log₇ 216 = log₇(8640/216) = log₇ 40.

Problem 26

If log_a x = v and log_a y = w, then log_a xy³ = ?

A) 3(v+w)
B) v + 3w
C) v + w
D) 3vw

Correct Answer: B

Split the product using the product property: log_a(xy³) = log_a x + log_a(y³).

Apply the power property to the second term: log_a(y³) = 3log_a y = 3w. Combined: v + 3w.

Problem 27

The total amount of money in a savings account at Bing's Bank is given by the formula A = A₀(2^t/3), where A is the total amount of money t years after an initial amount of A₀ was invested. Which of the following expressions gives the total number of years it will take for an initial investment of $50 to increase to $200?

A) log₃ 50
B) log₂ 25
C) 3log₅₀ 200
D) 3log₂ 4

Correct Answer: D

Substitute the given values: 200 = 50(2^t/3). Divide both sides by 50: 4 = 2^t/3.

Rewrite in logarithmic form: t/3 = log₂ 4. Multiply both sides by 3: t = 3log₂ 4.

Problem 28

If x is a positive integer, y is a negative integer, and log_5x z = y, what are the range of possible values of z?

A) z ≥ 1 and z ≤ −1
B) 0 < z ≤ 1
C) 0 < z ≤ 0.2
D) 0 < z ≤ 5

Correct Answer: C

Rewrite in exponential form: z = (5x)^y. Since y is a negative integer, this equals 1/(5x)^|y|, a positive fraction. Since x is a positive integer, 5x is at least 5, so z is always positive but less than 1.

The largest possible value of z occurs at the smallest base and smallest exponent magnitude: x=1 (so 5x=5) and |y|=1, giving z = 1/5 = 0.2. As x or |y| grows, z shrinks toward (but never reaches) 0. So the range is 0 < z ≤ 0.2.

Problem 29

If log_a x² = p and log_a(2y)³ = m, then log_a(2xy) = ?

A) 2p + 3m
B) 5pm
C) p/2 + m/3
D) pm/6

Correct Answer: C

From the given equations, isolate the single-power logs: log_a x = p/2 (dividing by the power property's exponent 2), and log_a(2y) = m/3 (dividing by exponent 3).

Split log_a(2xy) using the product property: log_a x + log_a(2y) = p/2 + m/3.

Problem 30

What is the solution of the equation log₄ [√(x+31) / (½x−1)] = 1?

A) −31
B) 3/4
C) 2
D) 5

Correct Answer: D

Rewrite in exponential form: √(x+31) / (½x−1) = 4¹ = 4, so √(x+31) = 4(½x−1) = 2x−4.

Square both sides: x+31 = (2x−4)² = 4x²−16x+16. Rearranging: 4x²−17x−15 = 0. Using the quadratic formula: x = (17 ± √(289+240))/8 = (17±23)/8, giving x = 5 or x = −0.75.

Since √(x+31) must be nonnegative, 2x−4 must also be nonnegative, requiring x ≥ 2. Only x = 5 satisfies this, so x = −0.75 is extraneous. Checking x=5: √36 = 6, and ½(5)−1 = 1.5, and 6/1.5 = 4 = 4¹ ✓.

Problem 31

For the function f(x) = 8^(x+1)/3, f(x) is defined for all x > 0. Which of the following expressions defines f⁻¹(x)?

A) 3log₈ x − 1
B) (1/3)log₈ x − 1
C) 3log₈(x−1)
D) 3/log₈x + 1

Correct Answer: A

Write the function as y = 8^(x+1)/3, then swap x and y to begin finding the inverse: x = 8^(y+1)/3.

Rewrite in logarithmic form: log₈ x = (y+1)/3. Multiply both sides by 3: 3log₈ x = y + 1. Subtract 1: y = 3log₈ x − 1, so f⁻¹(x) = 3log₈ x − 1.

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