Show that the solution of the equation $$5^x = 3^{x+4}$$ can be written as $$x = \frac{\ln 81}{\ln 5 - \ln 3}$$ Fully justify your answer. - AQA - A-Level Maths Mechanics - Question 6 - 2021 - Paper 2

Using the power rule of logarithms, we can rewrite this as:

$x \ln 5 = (x + 4) \ln 3$

We can rearrange this equation to isolate terms involving x:

$x \ln 5 - x \ln 3 = 4 \ln 3$

Factoring out x gives:

$x(\ln 5 - \ln 3) = 4 \ln 3$

Dividing both sides by (\ln 5 - \ln 3) leads us to:

$x = \frac{4 \ln 3}{\ln 5 - \ln 3}$

Furthermore, noting that (4 = \ln 81), we can express the solution as:

$x = \frac{\ln 81}{\ln 5 - \ln 3}$

Thus, we have shown the required result.