Let's go through the solution step by step.

Given:

$y = (9 + 8x)^{5/x}$

a. Find $\frac{dy}{dx}$ using logarithmic differentiation.

To use logarithmic differentiation, we start by taking the natural logarithm of both sides: $\ln y = \ln\left((9 + 8x)^{5/x}\right)$ Using the power rule for logarithms, we can bring the exponent $\frac{5}{x}$ down: $\ln y = \frac{5}{x} \ln(9 + 8x)$

Now, differentiate both sides with respect to $x$:

1. Differentiate the left side using implicit differentiation: $\frac{1}{y} \frac{dy}{dx}$
2. Differentiate the right side using the product rule: [ \frac{d}{dx}\left(\frac{5}{x} \ln(9 + 8x)\right) = \frac{5}{x} \cdot \frac{1}{9 + 8x} \cdot 8 - \frac{5 \ln(9 + 8