Answer
$\displaystyle \log\frac{x(x-2)}{15}$
Work Step by Step
$\log x+\log(x^{2}-4)-\log 15-\log(x+2)=$
... group all positive signed terms; group all negative signed terms
$=[\log x+\log(x^{2}-4)]-[\log 15+\log(x+2)]$
... apply the product rule to each bracket
$=\log(x(x^{2}-4))-\log(15(x+2))$
... apply the quotient rule
$=\displaystyle \log\frac{x(x^{2}-4)}{15(x+2)}$
... recognize a difference of squares in the numerator
$=\displaystyle \log\frac{x(x+2)(x-2)}{15(x+2)}$
... cancel the common term
$=\displaystyle \log\frac{x(x-2)}{15}$
