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ACT · Math · Preparing for Higher Math: Functions
Logarithms is part of Preparing for Higher Math: Functions in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.
Question types you will see here include evaluate a logarithm, convert between log and exponent form, product and quotient rules and power rule.
Common traps: dividing the argument by the base; ignoring a negative exponent for fractions; swapping the base and the argument.
Question 1
What is the value of log₆ 4 + log₆ 9?
log₆ 13
log₁₂ 36
2
36
Answer: C. 2
Adding logs with the same base multiplies the arguments: log₆(4 × 9) = log₆ 36 = 2. Choice A adds the arguments; B multiplies the arguments but also adds the bases; D finds the product 36 but forgets to take its logarithm.
Question 2
What is the value of log_2(8)?
3
4
5
6
Answer: A. 3
log_2(8) asks what power of 2 gives 8. Since 2^3 = 8, the value is 3.
Question 3
If log_x 36 = 2, what is the value of x?
−6
6
18
1,296
Answer: B. 6
x² = 36 gives x = 6 or −6, but a logarithm's base must be positive (and not 1), so x = 6. Choice A keeps the negative root; C divides 36 by 2; D squares 36 instead of taking its square root.
Question 4
What is the value of e^(2 ln 3)?
3
6
9
e⁶
Answer: C. 9
By the power rule, 2 ln 3 = ln 9, and e^(ln 9) = 9. Choice A drops the coefficient 2; B multiplies 2 × 3 instead of raising 3 to the power 2; D treats ln 3 as 3.
Question 5
If log₃ x = 4 and log₃ y = 2, what is the value of log₃(x²y)?
6
10
18
32
Answer: B. 10
log₃(x²y) = 2 log₃ x + log₃ y = 2(4) + 2 = 10. Choice A forgets the power rule (4 + 2); C squares the logarithm instead of doubling it (4² + 2); D multiplies the parts, 4² × 2.
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