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. $\ln(9)$
2. $\ln(4)$ (correct answer)
3. $\ln(15)$
4. $\ln\!\left(\dfrac{3}{12}\right)$

Explanation: This problem uses the logarithm quotient property. The quotient property states that $\ln(a) - \ln(b) = \ln(a/b)$. Applying this property to $\ln(12) - \ln(3)$, we get $\ln(12) - \ln(3) = \ln(12/3) = \ln(4)$. Choice A incorrectly keeps the arguments separate, while choice D incorrectly reverses the fraction.

Question 4

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 5

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 6

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

Explanation: This problem uses the property that natural logarithm and exponential are inverse functions. The key property is that ln(e^x) = x for any real number x. Applying this property directly to ln(e^5), we get ln(e^5) = 5. Choice A incorrectly leaves the expression unsimplified, while choice B incorrectly multiplies by e.

Question 7

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 8

1. $\dfrac{1}{2}$
2. $2$ (correct answer)
3. $7$
4. $49$

Explanation: This problem uses the fundamental relationship between logarithms and exponentials. The equation log_7(49) asks "to what power must we raise 7 to get 49?" Since 7^2 = 49, we have log_7(49) = 2. Choice A incorrectly gives the reciprocal, while choice C gives the base instead of the exponent.

Question 9

1. $2$
2. $3$ (correct answer)
3. $5$
4. $125$

Explanation: This problem uses the fundamental relationship between logarithms and exponentials. The equation log_5(125) asks "to what power must we raise 5 to get 125?" Since 5^3 = 125, we have log_5(125) = 3. Choice D incorrectly gives the argument of the logarithm rather than the exponent needed.

Question 10

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 11

1. $3^y = 27$ (correct answer)
2. $27^y = 3$
3. $3^3 = y$
4. $y^3 = 27$

Explanation: To convert from logarithmic to exponential form, we use the fundamental property that log_a(b) = c is equivalent to a^c = b. In the equation log₃(27) = y, the base is 3, the argument is 27, and the result is y. Converting to exponential form gives us 3^y = 27. Choice B incorrectly reverses the base and argument positions.

Question 12

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

Explanation: This problem uses the fundamental relationship between logarithms and exponentials. The property states that log_a(a^x) = x for any valid base a and exponent x. Since we have log(10^(-4)) and the base is 10, we can apply this property directly. Therefore, log(10^(-4)) = -4. Choice C incorrectly gives the argument rather than the logarithm value.

Question 13

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

Explanation: This problem uses the definition of logarithms with base 5. The expression $\log_5(125)$ asks "To what power must we raise 5 to get 125?" Since $5^3 = 5 \times 5 \times 5 = 125$, we have $\log_5(125) = 3$. Choice C would be incorrect as $5^5 = 3125$, which is much larger than 125.

Question 14

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 15

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 16

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 17

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 18

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 19

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 20

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 $x = 81$.