Find the value of $\sqrt[5]{0.00000165}$ given $\log165=2.2174839$ and $\log697424=5.8434968$
$\log x=\log\sqrt[5]{0.00000165}$
$\Rightarrow \log x =\dfrac{1}{5}\log0.00000165=\dfrac{1}{5}(\overline{6}.2174839$)
$\Rightarrow \dfrac{1}{5}(\overline{10}+4.2174839) = \overline{2}.8434968$
I don't understand how $\overline{2}.8434968$ came about. I can see the $\overline{2}$ came from $\dfrac{1}{5}\times\overline{10}$, but then where did the rest come from? I'm pretty sure this problem isn't asking me to use a calculator and plug in $\dfrac{1}{5} \times 4.2174839$. There's a trick I'm missing here. Thanks.
Since $\log 165 = 2.2174839$ is given, it means, the base is $10$. So we have $$ \log(0.00000165^{\frac{1}{5}}) = \frac{1}{5}\cdot \log(0.00000165) = \frac{1}{5}\cdot \log(\frac{165}{10^8}) \\ = \frac{1}{5}\cdot (\log(165) - \log(10^8)) = \frac{1}{5}\cdot (\underbrace{\log(165)}_{\text{given}} - 8\cdot\underbrace{\log(10)}_{=1}). $$