Writing as a single logarithm

Text

Write as a single logarithm the expression (with $x>1$) $2\log x+\log(x-1)-\tfrac12\log(x+3).$

Solution

Move the coefficients into exponents and use the sum and difference rules: $2\log x=\log x^{2},\qquad \tfrac12\log(x+3)=\log\sqrt{x+3}.$ Hence $\log x^{2}+\log(x-1)-\log\sqrt{x+3}=\log\frac{x^{2}(x-1)}{\sqrt{x+3}}.$ $\boxed{\;\log\dfrac{x^{2}(x-1)}{\sqrt{x+3}}\;}$