Find the exact solutions, in their simplest form, to the equations (a) $e^{x-9} = 8$ (b) $ ext{ln}(2y + 5) = 2 + ext{ln}(4 - y)$ - Edexcel - A-Level Maths Pure - Question 3 - 2017 - Paper 4

We start by isolating the logarithmic terms. We can express 2 as $\text{ln}(e^2)$, giving us:

$\text{ln}(2y + 5) = \text{ln}(e^2) + \text{ln}(4 - y)$

Using the property of logarithms that states $\text{ln}(a) + \text{ln}(b) = \text{ln}(ab)$, we rewrite the equation:

$\text{ln}(2y + 5) = \text{ln}(e^2(4 - y))$

Now, we exponentiate both sides:

$2y + 5 = e^2(4 - y)$

Next, we distribute $e^2$ on the right side:

$2y + 5 = 4e^2 - e^2y$

To isolate the terms involving $y$, we rearrange the equation:

$2y + e^2y = 4e^2 - 5$

Factor out $y$:

$y(2 + e^2) = 4e^2 - 5$

Finally, solve for $y$:

$y = \frac{4e^2 - 5}{2 + e^2}$