Answer
$\log_4{(\frac{x}{y}^{\frac{1}{3}}(x+1)^2)}$
Work Step by Step
The product rule for logarithms says that $\log_b{MN}=\log_bM+\log_bN$ i.e. the logarithm of a product is the sum of the logarithms.
The quotient rule for logarithms says that $\log_b{\frac{M}{N}}=\log_bM-\log_bN$ i.e. the logarithm of a quotient is the difference of the logarithms.
The power rule for logarithms says that $\log_b{M^p}=p\log_bM$ i.e. the logarithm of a number with an exponent is the exponent times the logarithm of the number.
$\log_ba=\frac{\log_ca}{\log_cb}$
Hence here: $\frac{1}{3}(\log_4 x-\log_4y)+2\log_4{(x+1)}=\frac{1}{3}\log_4{(\frac{x}{y})}+\log_4{(x+1)^2}=\log_4{(\frac{x}{y})^{\frac{1}{3}}}+\log_4{(x+1)^2}=\log_4{(\frac{x}{y}^{\frac{1}{3}}(x+1)^2)}$
