How do you expand this logarithm? log_(4)sqrt(x^3)

${\log}_{4} \sqrt{{x}^{3}}$

Feb 12, 2017

${\log}_{4} \sqrt[3]{x} = \frac{1}{3} {\log}_{4} x = \frac{1}{3} \log \frac{x}{\log} 4 = 0.5537 \log x$

Explanation:

As #log_aroot(n)x=log_a(x)^(1/n)=1/nlog_ax,

As such ${\log}_{4} \sqrt[3]{x} = {\log}_{4} {\left(x\right)}^{\frac{1}{3}} = \frac{1}{3} {\log}_{4} x$

but as ${\log}_{a} b = \log \frac{b}{\log} a$

${\log}_{4} \sqrt[3]{x} = \frac{1}{3} \log \frac{x}{\log} 4 = \frac{1}{3} \log \frac{x}{0.6021} = 0.5537 x$