ACT Math Quiz: Logarithmic Functions

What this quiz covers

This quiz focuses on Logarithmic Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $5$
2. $7$ (correct answer)
3. $9$
4. $11$

Explanation: This is a logarithms question testing the product rule and extraneous solution detection. Choice B (7) is correct — apply the log product rule: log₃(x + 2) + log₃(x − 4) = log₃((x + 2)(x − 4)) = 3. Convert to exponential form: (x + 2)(x − 4) = 3³ = 27. Expand: x² − 2x − 8 = 27 → x² − 2x − 35 = 0 → (x − 7)(x + 5) = 0 → x = 7 or x = −5. Check: x = −5 makes log₃(−5 + 2) = log₃(−3), which is undefined (can't take log of a negative). So x = 7 is the only valid solution. Choice A (5) comes from a factoring error: perhaps solving x² − 2x − 35 = 0 as (x − 5)(x + 7) = 0. Choice C (9) comes from treating each log separately: log₃(x + 2) = 3 → x + 2 = 27 → x = 25... or log₃(x − 4) = 3 → x − 4 = 27 → x = 31. Choice D (11) comes from adding: (x + 2) + (x − 4) = 27 → 2x − 2 = 27 → x = 14.5, rounding or computing differently. Pro tip: After applying the log product rule, you'll get a quadratic. It will typically have two roots — always check BOTH in the original equation. A root that produces a negative or zero argument for any logarithm is extraneous and must be discarded.

Question 2

1. $\log(52)$
2. $\log\!\left(\dfrac{2}{50}\right)$
3. $\log(100)$ (correct answer)
4. $\log(2\cdot 50)\cdot\log(10)$

Explanation: This problem uses the logarithm product property. The product property states that log_a(x) + log_a(y) = log_a(xy). Applying this property to log(2) + log(50), we get log(2) + log(50) = log(2 × 50) = log(100). Choice A incorrectly adds the arguments instead of multiplying them.

Question 3

1. 3 (correct answer)
2. 2
3. 10
4. 1

Explanation: To evaluate this logarithm, we need to find what power 10 must be raised to get 1000. The logarithm property states that log_a(b) = c means a^c = b. We can rewrite 1000 as 10^3, so log₁₀(1000) = log₁₀($10^3$). Using the power rule for logarithms, log_a($x^n$) = n·log_a(x), we get 3·log₁₀(10) = 3·1 = 3.

Question 4

1. 4 (correct answer)
2. 3
3. 2
4. 5

Explanation: To evaluate this logarithm, we need to find what power 3 must be raised to get 81. The logarithm property states that $\log_a(b) = c$ means $a^c = b$. We can rewrite 81 as $3^4$ (since $3 \cdot 3 \cdot 3 \cdot 3 = 81$). Therefore, $\log_3(81) = \log_3(3^4)$. Using the power rule, this equals $4 \cdot \log_3(3) = 4 \cdot 1 = 4$.

Question 5

1. $-2$
2. $10^{-2}$ (correct answer)
3. $2$
4. $-10^{2}$

Explanation: Given log(x) = -2 (base 10 implied), we need to find x. Using the definition log₁₀(x) = -2 means 10^(-2) = x. Therefore, x = 10^(-2) = 1/10² = 1/100 = 0.01. Choice A gives just -2, which is the logarithm value, not x itself.

Question 6

1. 100 (correct answer)
2. 10
3. 20
4. 200

Explanation: To solve this equation, we need to convert from logarithmic to exponential form. The equation log(x) = 2 means "10 raised to what power equals x?" Since log without a specified base typically means log₁₀, we have 10² = x. Therefore, x = 100. Choice B would give 10¹ = 10, which doesn't satisfy the original equation.

Question 7

1. $0$
2. $1$ (correct answer)
3. $e$
4. $-1$

Explanation: This problem uses the fundamental property that ln(1) = 0 and the inverse relationship between natural logarithm and exponential functions. The equation ln(x) = 0 means that e^0 = x. Since e^0 = 1 for any base, we have x = 1. Choice A incorrectly gives the exponent value, while choice C gives the base of natural logarithm.

Question 8

1. 4
2. 8 (correct answer)
3. 16
4. 32

Explanation: This is a logarithms question testing the product rule. Choice B (8) is correct — apply the log product rule: log₂(x) + log₂(4) = log₂(4x) = 5. Convert to exponential form: 4x = 2⁵ = 32. Solve: x = 8. Since log₂(4) = 2, you can also solve as: log₂(x) = 5 − 2 = 3 → x = 2³ = 8. Choice A (4) results from computing 2³ incorrectly — arriving at the right exponent of 3 but evaluating 2³ as 4 (possibly confusing 2² = 4 with 2³ = 8). Choice C (16) results from an off-by-one exponent error after correctly applying the product rule: correctly getting log₂(x) = 3, but then computing x = 2⁴ = 16 instead of 2³ = 8. Choice D (32) ignores the log₂(4) term entirely, solving log₂(x) = 5 → x = 2⁵ = 32. Pro tip: The log product rule states log_b(M) + log_b(N) = log_b(MN). Use it to combine the two log terms before converting to exponential form — this is almost always faster than working with them separately.

Question 9

1. $2$ (correct answer)
2. $1$
3. $e$
4. $e^2$

Explanation: To evaluate this natural logarithm, we use the power rule for logarithms. The property states that $\log_a(x^n) = n \cdot \log_a(x)$. Since ln means log_e, we have $\ln(e^2)$. Using the power rule: $\ln(e^2) = 2 \cdot \ln(e) = 2 \cdot 1 = 2$. This demonstrates that the natural logarithm and exponential functions with base e are inverse operations.

Question 10

1. $2^{32}=5$
2. $5^{2}=32$
3. $2^{5}=32$ (correct answer)
4. $32^{5}=2$

Explanation: To convert log₂(32) = 5 to exponential form, we use the definition that log_a(b) = c means a^c = b. Here, a = 2, b = 32, and c = 5. Therefore, the exponential form is 2⁵ = 32. We can verify: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Choice A incorrectly swaps the base and result.

Question 11

1. $2^{100} = 10$
2. $2^{10} = 100$
3. $10^3 = 100$
4. $10^2 = 100$ (correct answer)

Explanation: To convert from logarithmic to exponential form, we use the fundamental property that log_a(b) = c is equivalent to a^c = b. Given log₁₀(100) = 2, the base is 10, the exponent is 2, and the result is 100. Therefore, the equivalent exponential equation is 10^2 = 100. Choice B incorrectly switches the base and exponent positions.

Question 12

1. $\log(x) + \log(y)$ (correct answer)
2. $\log(x) \cdot \log(y)$
3. $\frac{\log(x)}{\log(y)}$
4. $\log(x) - \log(y)$

Explanation: This problem requires the product rule for logarithms. The logarithm property for products states that $\log_a(xy) = \log_a(x) + \log_a(y)$. This means when we have the logarithm of a product, we can separate it into the sum of individual logarithms. Therefore, $\log(xy) = \log(x) + \log(y)$. Choice B incorrectly suggests multiplying logarithms instead of adding them.

Question 13

1. 1
2. 3
3. 2 (correct answer)
4. 4

Explanation: To evaluate this logarithm, we use the definition: $\log_7(49)$ asks "To what power must we raise 7 to get 49?" Since $7^2$ = $7 \times 7$ = 49, we have $\log_7(49)$ = 2. Choice B would be incorrect because $7^3$ = 343, not 49.

Question 14

1. 81 (correct answer)
2. 12
3. 9
4. 3

Explanation: To solve this equation, we need to convert from logarithmic to exponential form. The logarithm property states that $log_a(b) = c$ means $a^c = b$. Given $log_3(x) = 4$, we convert to exponential form: $3^4 = x$. Calculating $3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81$, so $x = 81$.

Question 15

1. $\log(y) - \log(x)$
2. $\log(x) + \log(y)$
3. $\log(x) \cdot \log(y)$
4. $\log(x) - \log(y)$ (correct answer)

Explanation: This problem uses the quotient rule for logarithms. The quotient rule states that log(x/y) = log(x) - log(y). This property allows us to separate the logarithm of a fraction into the difference of two logarithms. Choice B incorrectly suggests adding the logarithms, which would apply to a product, not a quotient.

Question 16

1. $3$
2. $2$
3. $\dfrac{1}{2}$ (correct answer)
4. $\dfrac{1}{3}$

Explanation: This problem uses the relationship between logarithms with different bases that are powers of each other. Since 9 = 3^2, we need to find what power of 9 equals 3. This means 9^x = 3, or (3^2)^x = 3^1, which gives us 3^(2x) = 3^1. Therefore, 2x = 1, so x = 1/2. Choice B incorrectly gives the reciprocal relationship.

Question 17

1. $\log(x)+\log(y)$ (correct answer)
2. $\log(x)-\log(y)$
3. $\log(x)\cdot\log(y)$
4. $\log\!\left(\dfrac{x}{y}\right)$

Explanation: We need the product rule for logarithms to simplify log(xy). The product rule states that log_a(xy) = log_a(x) + log_a(y) for positive x and y. Therefore, log(xy) = log(x) + log(y). Choice C incorrectly multiplies the logarithms, while choice B would apply to log(x/y).

Question 18

1. $e^5$
2. $5e$
3. $5$ (correct answer)
4. $\ln(5e)$

Explanation: To evaluate ln(e⁵), we use the property that logarithms and exponentials are inverse functions. Specifically, log_a($a^x$) = x for any valid base a. Since ln is the natural logarithm with base e, we have ln(e⁵) = 5. This follows directly from the definition: if ln(e⁵) = x, then e^x = e⁵, so x = 5.

Question 19

1. $15$
2. $125$ (correct answer)
3. $\dfrac{3}{5}$
4. $5^{1/3}$

Explanation: To solve log₅(x) = 3, we use the definition that log_a(b) = c means a^c = b. Here, log₅(x) = 3 means 5³ = x. Therefore, x = 5³ = 5 × 5 × 5 = 125. Choice C incorrectly gives 3/5, while choice D would be the solution to log₅(x) = 1/3.

Question 20

1. $\ln(x)+\ln(y)$
2. $\ln(x)-\ln(y)$ (correct answer)
3. $\dfrac{\ln(x)}{\ln(y)}$
4. $\ln(xy)$

Explanation: To express ln(x/y) in terms of ln(x) and ln(y), we use the quotient rule for logarithms. The quotient rule states that log_a(x/y) = log_a(x) - log_a(y). Since this applies to natural logarithms as well, ln(x/y) = ln(x) - ln(y). Choice A would be correct for ln(xy), not ln(x/y).