CAT 2020 Slot 1 — QA Question 25

Write the given equation and isolate $\log_4 y$.

$$\begin{aligned} &\log_4 5=(\log_4 y)(\log_6 \sqrt5) \quad\text{(given)}\\ &\Rightarrow\ \frac{\log_4 5}{\log_4 y}=\log_6 \sqrt5 \end{aligned}$$

Change base on the left. The ratio $\dfrac{\log_4 5}{\log_4 y}$ is just $\log_y 5$.

$$\begin{aligned} &\log_y 5=\log_6 \sqrt5 \quad\text{(change of base)}\\ &\Rightarrow\ \log_y 5=\tfrac12\,\log_6 5 \quad\text{(since $\sqrt5=5^{1/2}$)} \end{aligned}$$

Fold the $\tfrac12$ into the base. Using $\tfrac12\log_6 5=\log_{6^2}5$, the bases must match.

$$\begin{aligned} &\log_y 5=\log_{36} 5 \quad\text{($\tfrac12\log_6 5=\log_{36}5$)}\\ &\Rightarrow\ y=36 \end{aligned}$$