Question

What is 3 log Subscript 2 Baseline x minus (log Subscript 2 Baseline 3 minus log Subscript 2 Baseline (x + 4)) written as a single logarithm?
log Subscript 2 Baseline left-bracket StartFraction x cubed Over (StartFraction 3 Over x + 4 EndFraction) EndFraction Right-bracket
log Subscript 2 Baseline (StartFraction 3 x cubed Over x + 4 EndFraction)
log Subscript 2 Baseline left-bracket (StartStartFraction x cubed Over 3 EndFraction) Over x + 4 EndEndFraction Right-bracket
log Subscript 2 Baseline (StartFraction x cubed Over 3 + (x + 4) EndFraction)

Answer 1

Start with 3 log_2 x - (log_2 3 - log_2(x+4))
= 3 log_2 x - log_2 3 + log_2(x+4)
= log_2(x^3) + log_2(x+4) - log_2 3
= log_2( x^3(x+4) / 3 ).

So the single logarithm is log_2( x^3(x+4) / 3 ). (Domain: x > 0.)