2log(x + a) = log(16a^y), where a is a positive constant Find x in terms of a, giving your answer in its simplest form - Edexcel - A-Level Maths Pure - Question 9 - 2016 - Paper 2

We start with the equation:

$\log(9y + b) - \log(2y - b) = 2$

Applying the quotient rule, we rewrite this as:

$\log\left(\frac{9y + b}{2y - b}\right) = 2$

Exponentiating both sides leads to:

$\frac{9y + b}{2y - b} = 10^2 = 100$

From here, we can multiply across to eliminate the fraction:

$9y + b = 100(2y - b)$

Expanding this gives:

$9y + b = 200y - 100b$

Next, we rearrange to isolate terms involving $y$:

$9y + 100b = 200y$

Thus,

$100b = 200y - 9y$

Combining like terms yields:

$y = \frac{100b}{191}$