Question

Find the EXACT solution to the exponential equation 500=20(1.5)^(4x)

Answer 1

$500=20(1.5)^{4x}\implies \cfrac{500}{20}=1.5^{4x}\implies 25=1.5^{4x} \\\\\\ \textit{now, we'll take a common log, any will do, we'll use ln()} \\\\\\ ln(25)=ln\left(1.5^{4x} \right)\\\\ -----------------------------\\\\ log_{{ a}}\left( x^{{ b}} \right)\implies {{ b}}\cdot log_{{ a}}(x)\\\\ -----------------------------\\\\ thus\implies ln(25)=4xln(1.5)\implies \cfrac{ln(25)}{4ln(1.5)}=x$