Question

Consider the equation below.

log Subscript 4 Baseline (x + 3) = log Subscript 2 Baseline (2 + x)

Which system of equations can represent the equation?
y 1 = StartFraction log (x + 3) Over log 4 EndFraction, y 2 = StartFraction log (2 + x) Over log 2 EndFraction
y 1 = StartFraction log x + 3 Over log 4 EndFraction, y 2 = StartFraction log 2 + x Over log 2 EndFraction
y 1 = StartFraction log 4 Over log 2 EndFraction, y 2 = StartFraction log (x + 3) Over log (2 + x) EndFraction
y 1 = StartFraction log x + 3 Over 4 EndFraction, y 2 = StartFraction log 2 + x Over 2 EndFraction

Answer 1

The first system.

Using change of base, log_4(x+3) = ln(x+3)/ln4 and log_2(2+x) = ln(2+x)/ln2, so
y1 = log(x+3)/log4 and y2 = log(2+x)/log2.
(Also require x+3>0 and 2+x>0 ⇒ x>-2.)