Single correct option
If log_a b = 2 and log_b c = 3, where a, b, c > 1, what is the value of log_(ab) (bc)?
- A. 8/3
- B. 3
- C. 5/2
- D. 2
Show answer and solution
Answer: (A) 8/3
From log_a b = 2, b = a². From log_b c = 3, c = b³ = a⁶.
Then ab = a · a² = a³ and bc = a² · a⁶ = a⁸.
log_(ab)(bc) = log_(a³)(a⁸) = 8/3.
Why others fail: 3 is just log_b c; 5/2 comes from adding the two given logs and halving; 2 is log_a b.
Tip: Express every variable as a power of a single base – the log expression then collapses to a ratio of exponents.
Then ab = a · a² = a³ and bc = a² · a⁶ = a⁸.
log_(ab)(bc) = log_(a³)(a⁸) = 8/3.
Why others fail: 3 is just log_b c; 5/2 comes from adding the two given logs and halving; 2 is log_a b.
Tip: Express every variable as a power of a single base – the log expression then collapses to a ratio of exponents.
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