How do you calculate  log_3 sqrt243?

Apr 24, 2016

First, use the exponent rule $\sqrt[a]{n} = {n}^{\frac{1}{a}}$

Explanation:

${\log}_{3} \left({243}^{\frac{1}{2}}\right)$

Now, use the log rule $\log {a}^{n} = n \log a$

$\frac{1}{2} {\log}_{3} \left(243\right)$

Now, use the log rule ${\log}_{a} n = \log \frac{n}{\log} a$

$\frac{1}{2} \left(\log \frac{243}{\log} 3\right)$

Rewrite 243 in base 3.

$\frac{1}{2} \left(\frac{\log {3}^{5}}{\log} 3\right)$

$\frac{1}{2} \left(\frac{5 \log 3}{\log} 3\right)$

$\frac{1}{2} \times 5$

$\frac{5}{2}$

Thus, completely simplified, ${\log}_{3} \left(\sqrt{243}\right) = \frac{5}{2} \mathmr{and} 2.5$

Hopefully this helps!