Question

Fully condense the expression e=" 2\log_5x-\log_52+\log_5x " alt=" 2\log_5x-\log_52+\log_5x " align="absmiddle" class="latex-formula">
In a sentence or two, describe the properties you used.

Answer 1

$\bf \textit{logarithm of factors} \\\\ log_a(xy)\implies log_a(x)+log_a(y) \\\\\\ \textit{Logarithm of rationals} \\\\ log_a\left( \frac{x}{y}\right)\implies log_a(x)-log_a(y) \\\\\\ \textit{Logarithm of exponentials} \\\\ log_a\left( x^b \right)\implies b\cdot log_a(x)\\\\ -------------------------------$

$\bf 2log_5(x)-log_5(2)+log_5(x)\implies {\stackrel{\textit{of exponentials}}{log_5(x^2)}-log_5(2)}+log_5(x) \\\\\\ \stackrel{\textit{\textit{of rationals}}}{log_5\left( \cfrac{x^2}{2} \right)}+log_5(x)\implies \stackrel{\textit{of factors}}{log_5\left(\cfrac{x^2}{2}\cdot x \right)}\implies log_5\left( \cfrac{x^3}{2} \right)$