Problem 2 · Translate logarithms into exponents · 3 points
Made by AI and not reviewed by a real person.
For x > 0 with x ≠ 1, suppose logₓ 16 = log₂ x. What is the sum of all possible values of x?
- 1
- 4
- 0
- 17/4
- 1/4
Show a hint
- Rewrite each logarithm as an exponential statement. Remember that a logarithm’s base can lie between 0 and 1.
- Let t = log₂ x. Then x = 2^t, and the given equation also says x^t = 16.
Show the answer and solution
Answer: D — 17/4
Let t = log₂ x. Since x = 2^t and x^t = 16, we have 2^(t²) = 2⁴. Thus t = ±2, giving x = 4 or x = 1/4. Their sum is 17/4. The tempting answer 4 misses the valid base 1/4, for which both logarithms are negative.