2740.38 – Find k (logarithms)

Start with $x = 10^k \rightarrow 10 = x^{\small \left(\dfrac{1}{k}\right)}$. Then $log_x(10) = \dfrac{1}{k}$.

For $log_{x^2}(10)$ Observe that $(x^2)^{\small \left(\dfrac{1}{2k}\right)} = x^{\small \left(\dfrac{1}{k}\right)} = 10$, so $log_{x^2}(10) = \dfrac{1}{2k}$.

So $log_x(10) + log_{x^2}(10) = \dfrac{1}{k} + \dfrac {1}{2k} = 10$, as the problem tells us. Therefore:

$$\begin{align*}\dfrac{2}{2k} + \dfrac{1}{2k} &= 10 \\[1em]\dfrac{3}{2k} &= 10 \\[1em]2k &= \dfrac{3}{10} \\[1em]k &= \dfrac{3}{20}\end{align*}$$