Question 1
What is the value of log_2(8)?
- A.
3
- B.
4
- C.
5
- D.
6
Answer: A. 3
log_2(8) asks what power of 2 gives 8. Since 2^3 = 8, the value is 3.
ACT · Math · Preparing for Higher Math: Functions
Logarithms
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.
Sample questions
Question 2
Which of the following is equivalent to the equation 4^(x + 1) = 50?
- A.
log₄(x + 1) = 50
- B.
x + 1 = log₅₀ 4
- C.
x = log₄ 50 + 1
- D.
x + 1 = log₄ 50
Answer: D. x + 1 = log₄ 50
An exponential equation b^E = R becomes E = log_b R, so x + 1 = log₄ 50. Choice A puts the exponent inside the log and makes 50 the result; B swaps the base and the result; C moves the 1 to the wrong side, adding where it should subtract.
- A.
Question 3
If log₂ x = 3, what is the value of log₂ (4x³) ?
- A.
11
- B.
15
- C.
29
- D.
108
Answer: A. 11
log₂(4x³) = log₂ 4 + 3 log₂ x = 2 + 3(3) = 11. Choice B cubes the 4 as well, 3(2 + 3); C cubes the log value, 2 + 3³; D substitutes x = 3 and computes 4 · 3³, confusing x with log₂ x.
- A.
Question 4
What are all the solutions of log₃(x² − 8x) = 2?
- A.
−9 and 1
- B.
9 only
- C.
4 − √22 and 4 + √22
- D.
−1 and 9
Answer: D. −1 and 9
Rewrite as x² − 8x = 9, so (x − 9)(x + 1) = 0. Both x = 9 and x = −1 make x² − 8x = 9 > 0, so both are valid; the argument, not x itself, must be positive. Choice A has the factor signs reversed; B rejects −1 just because it is negative; C sets x² − 8x = 3 × 2 = 6.
- A.
Question 5
What is the sum of all solutions of e^(2x) − 5eˣ + 6 = 0?
- A.
ln 5
- B.
ln 6
- C.
5
- D.
6
Answer: B. ln 6
Let u = eˣ: u² − 5u + 6 = 0 gives u = 2 or 3, so x = ln 2 or ln 3. Their sum is ln 2 + ln 3 = ln 6. Choice A takes ln of the sum of the u-values; C adds the u-values and stops; D multiplies the u-values and never takes ln.
- A.
Find your real gaps
A few questions show the style. The free scan shows which ACT skills cost you the most points, then builds a plan around them.