Question

What is 2log5 (5x^3) + 1/3 log5 (x^2 + 6) written as a single logarithm?

Answer 1

`\begin{array}{llll} \textit{Logarithm of exponentials} \\\\ \log_a\left( x^b \right)\implies b\cdot \log_a(x) \end{array} ~\hspace{7em} \begin{array}{llll} \textit{logarithm of factors} \\\\ \log_a(xy)\implies \log_a(x)+\log_a(y) \end{array} \\\\[-0.35em] ~\dotfill`

`2\log_5(5x^3)+\cfrac{1}{3}\log_5(x^2+6)\implies \log_5( ~~ (5x^3)^2 ~~ )+\log_5\left( ~~ (x^2+6)^{(1)/(3)} ~~ \right) \\\\\\ \log_5( ~~ 25x^6 ~~ )+\log_5\left( ~~ (x^2+6)^{(1)/(3)} ~~ \right)\implies \log_5\left( ~~ (25x^6)\sqrt[3]{(x^2+6)} ~~ \right)`