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 Pure - Question 6 - 2021 - Paper 2

Next, we can expand the right side:

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

Now, we'll isolate the terms involving x:

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

Factoring x out from the left-hand side, we have:

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

Finally, we solve for x:

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

To express this in the form required, we recognize that:

$4 \ln 3 = \ln(3^4) = \ln(81)$

Thus, the solution can be written as:

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

This completes our proof.