Problem 5 · Symmetrize reciprocal log products · 4 points
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Let x > 1 satisfy x + 1/x = 3, and let log mean base 10. Which single logarithm equals log(x³ − 1) − log(x − 1) + log(1 + x⁻³) − log(1 + x⁻¹)?
- log 8
- log 9
- log 7
- log(1/8)
- log 3
Show a hint
- Try to make the expression symmetric in x and 1/x before solving for x.
- Combine the logarithms, then factor the cubic expressions and look for x² + 1/x².
Show the answer and solution
Answer: A — log 8
Combining the logarithms gives log(((x³ − 1)(1 + x⁻³))/((x − 1)(1 + x⁻¹))). Factoring and cancelling reduces the argument to ((x² + x + 1)(x² − x + 1))/x² = x² + 1 + x⁻². This equals (x + 1/x)² − 1 = 8, so the answer is log 8. The tempting log 9 comes from overlooking the −1 in that final identity.