2740.38 – Find k (logarithms)

Start with $x = 10^k \rightarrow 10 = x^\frac{1}{k}$. Then $log_x10 = \frac{1}{k}$.

Now, $log_x^2 10: log_x10 = \frac{1}{k} \rightarrow x^\frac{1}{k} = 10 \rightarrow (x^2)^\frac{1}{2k} = 10 \rightarrow log_x^2 10 = \frac{1}{2k}.$

So $log_x10 + log_x^210 = \frac{1}{k} + \frac {1}{2k} = 10$ (given)

$\rightarrow \frac{2}{2k} + \frac{1}{2k} = 10 \rightarrow \frac{3}{2k} = 10 \rightarrow 2k = \frac{3}{10} \rightarrow k = \frac{3}{20}.$