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ACT Math Help: Logarithmic Functions

Review real example questions for Logarithmic Functions in ACT Math.

Question 1 / 10

0 of 10 answered

What is the value of $x$ that satisfies $\log_3(x + 2) + \log_3(x - 4) = 3$ ?

All questions

Question 1

What is the value of $x$ that satisfies $\log_3(x + 2) + \log_3(x - 4) = 3$ ?

$5$
$7$ (correct answer)
$9$
$11$

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

A worksheet asks you to simplify $\log(2)+\log(50)$ (base 10). Which single logarithm is equivalent?

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

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

What is $\log_{10}(1000)$ ?

3 (correct answer)
2
10
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

Evaluate $\log_3(81)$ .

4 (correct answer)
3
2
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

A calculator app uses base-10 logs. If $\log(x)= -2$ , what is the value of $x$ ?

$-2$
$10^{-2}$ (correct answer)
$2$
$-10^{2}$

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

What is the value of $x$ if $\log(x) = 2$ ?

100 (correct answer)
10
20
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

A student solves the equation $\ln(x)=0$ . What is the value of $x$ ?

$0$
$1$ (correct answer)
$e$
$-1$

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

If $\log_2(x) + \log_2(4) = 5$ , what is the value of $x$ ?

4
8 (correct answer)
16
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

What is $\ln(e^2)$ ?

$2$ (correct answer)
$1$
$e$
$e^2$

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

A student is rewriting a logarithmic equation in exponential form. Which equation is equivalent to $\log_{2}(32)=5$ ?

$2^{32}=5$
$5^{2}=32$
$2^{5}=32$ (correct answer)
$32^{5}=2$

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.