Question

What is 2 log Subscript 5 Baseline (5 x cubed) + one-third log Subscript 5 Baseline (x squared + 6) written as a single logarithm?
log Subscript 5 Baseline (25 x Superscript 5 Baseline) RootIndex 3 StartRoot x squared + 6 EndRot
log Subscript 5 Baseline (25 x Superscript 6 Baseline) RootIndex 3 StartRoot x squared + 6 EndRoot
log Subscript 5 Baseline 25 x Superscript 5 Baseline + log Subscript 5 Baseline 2 + log one-third x squared
(log Subscript 5 Baseline 25 + log Subscript 5 Baseline Superscript 6 Baseline) (log Subscript 5 Baseline RootIndex 3 StartRoot x squared + 6 EndRoot)

Answer 1

Use log rules: a log_b(A) + c log_b(B) = log_b(A^a B^c).

2 log_5(5x^3) + (1/3) log_5(x^2+6)
= log_5((5x^3)^2 (x^2+6)^{1/3})
= log_5(25 x^6 (x^2+6)^{1/3}).

(Requirement: 5x^3>0 so x>0.)