Question

Combine into a single logarithm

Answer 1

`\displaystyle =\log\biggr(((x+y)^3(x-y)^2)/(x^2+y^2)\biggr)`

Explanation:

`\displaystyle 3\log(x+y)+2\log(x-y)-\log(x^2+y^2)\\\\=\log((x+y)^3)+\log((x-y)^2)-\log(x^2+y^2)\\\\=\log\biggr((x+y)^3}{(x-y)^2}\biggr)-\log(x^2+y^2)\\\\=\log\biggr(((x+y)^3(x-y)^2)/(x^2+y^2)\biggr)`

Answer 2

`\textit{Logarithm of exponentials} \\\\ \log_a\left( x^b \right)\implies b\cdot \log_a(x) \\\\\\ \begin{array}{llll} \textit{logarithm of factors} \\\\ \log_a(xy)\implies \log_a(x)+\log_a(y) \end{array} ~\hspace{4em} \begin{array}{llll} \textit{Logarithm of rationals} \\\\ \log_a\left( (x)/(y)\right)\implies \log_a(x)-\log_a(y) \end{array} \\\\[-0.35em] ~\dotfill`

`3\log(x+y)+2\log(x-y)-\log(x^2+y^2) \\\\\\ \log[(x+y)^3]+\log[(x-y)^2]-\log(x^2+y^2) \\\\\\ \log[(x+y)^3(x-y)^2]-\log(x^2+y^2)\implies \log\left[ \cfrac{(x+y)^3(x-y)^2}{x^2+y^2} \right]`