Question

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

Answer 1

`\bf \textit{logarithm of factors} \\\\ log_a(xy)\implies log_a(x)+log_a(y) \\\\\\ \textit{Logarithm of exponentials} \\\\ log_a\left( x^b \right)\implies b\cdot log_a(x)\\\\ -------------------------------\\\\ 2log_5(5x^3)+\cfrac{1}{3}log_5(x^2+6)\implies log_5[~~(5x^3)^2~~]+log_5[~~(x^2+6)^{(1)/(3)}~~] \\\\\\ log[~(5^2x^(3\cdot 2))~]+log_5[~\sqrt[3]{x^2+6}~]\implies log[~(25x^6)~]+log_5[~\sqrt[3]{x^2+6}~] \\\\\\ log_5[~~(25x^6)(\sqrt[3]{x^2+6})~~]\implies log_5(25x^6\sqrt[3]{x^2+6})`