Question

Find log_{3} [27*\sqrt[4]{3} ] wihout using a calculator. Please show all work!

Answer 1

$\bf \textit{logarithm of factors} \\\\ log_a(xy)\implies log_a(x)+log_a(y)\\\\ -------------------------------\\\\ log_3(27\sqrt[3]{3})\qquad \begin{cases} 27=3\cdot 3\cdot 3\\ \qquad 3^3\\ \sqrt[3]{3}=\sqrt[3]{3^1}\\ \qquad 3^{\frac{1}{3}} \end{cases}\implies log_3\left( 3^3\cdot 3^{\frac{1}{3}} \right) \\\\\\ log_3\left( 3^{3+\frac{1}{3}} \right)\implies log_3\left( 3^{\frac{4}{3}} \right)\implies \cfrac{4}{3}\cdot log_3(3)\implies \cfrac{4}{3}\cdot 1\implies \cfrac{4}{3}$