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 4 - 2017 - Paper 4

To solve for $y$, first rewrite the equation:

$ext{ln}(2y + 5) - ext{ln}(4 - y) = 2$

Using the property of logarithms, combine the left side:

$ext{ln}\left(\frac{2y + 5}{4 - y}\right) = 2$

Now exponentiate both sides to eliminate the logarithm:

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

Next, cross-multiply to solve for $y$:

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

Expanding this gives:

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

Combining like terms:

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

Factoring out $y$ from the left side:

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

Finally, isolate $y$:

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

Thus, the exact solution for part (b) is:

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