Solve: log₈(x³)/log₈(x^1.5) + log₇(x⁵)/log₄₉(x) Logarithmic Expression

To solve the given problem, we begin by simplifying each component of the expression.

Step 1: Simplify $\frac{\log_8x^3}{\log_8x^{1.5}}$.
Applying the power rule of logarithms, we get:
$\log_8x^3 = 3 \log_8x$, and $\log_8x^{1.5} = 1.5 \log_8x$.
Thus, $\frac{3 \log_8x}{1.5 \log_8x} = \frac{3}{1.5} = 2$.

Step 2: Simplify $\frac{1}{\log_{49}x} \times \log_7x^5$.
First, notice that $\log_7x^5 = 5 \log_7x$ by the power rule.
Applying the change of base formula, $\log_{49}x = \frac{\log_7x}{\log_749} = \frac{\log_7x}{2}$ because $49 = 7^2$.
This gives $\frac{1}{\log_{49}x} = \frac{2}{\log_7x}$.
Therefore, $\frac{2}{\log_7x} \times 5 \log_7x = 2 \times 5 = 10$.

Step 3: Combine the results from Step 1 and Step 2.
The simplified expression is $2 + 10 = 12$.

Therefore, the solution to the problem is $12$.