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

First, isolate the logarithm:

$$\ln(2y + 5) - \ln(4 - y) = 2$$

Using the property of logarithms that states (\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)), we can rewrite the equation as:

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

Next, exponentiate both sides to remove the logarithm:

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

Cross-multiply to solve for (y):

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

Expanding the right side gives:

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

Now, collect all terms involving (y) on one side:

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

Factor out (y):

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

Finally, isolate (y):

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

This is the exact solution.