Simplify the Product: 1/(ln 4) × 1/(log₈ 10) Logarithmic Expression

$\frac{1}{\ln4}\cdot\frac{1}{\log_810}=$

To solve the problem, we must evaluate the expression $\frac{1}{\ln 4} \cdot \frac{1}{\log_8 10}$.

First, convert $\log_8 10$ using the change of base formula. We have:

• $\log_8 10 = \frac{\ln 10}{\ln 8}$.

Substitute this back into the original expression:

$\frac{1}{\ln 4} \cdot \frac{1}{\log_8 10} = \frac{1}{\ln 4} \cdot \frac{\ln 8}{\ln 10}$.

Next, we need to simplify the expression. We know that $\ln 8 = \ln (2^3) = 3 \ln 2$ and $\ln 4 = \ln (2^2) = 2 \ln 2$.

Substitute these into the expression:

= $\frac{1}{2 \ln 2} \cdot \frac{3 \ln 2}{\ln 10}$.

Simplify by canceling $\ln 2$:

= $\frac{3}{2} \cdot \frac{1}{\ln 10}$.

Now express $\ln 10 = \ln (e \cdot \log e)$, meaning this is equivalent to $\log e$. Continuing, the expression $\frac{3}{2} \cdot \frac{1}{\log e} = \frac{3}{2} \log e$.

Therefore, the simplified solution to the given expression is $\frac{3}{2} \log e$.