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

Using the property of logarithms, which states that $\ln(a^b) = b\ln(a)$, we can rewrite the equation as:

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

Next, expand the right side and rearrange to isolate terms involving $x$:

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

This leads to:

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

Factoring out $x$ gives us:

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

Finally, solve for $x$ by dividing both sides by $(\ln(5) - \ln(3))$:

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

Recognizing that $\ln(81) = 4 \ln(3)$, we substitute to obtain:

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

This demonstrates that the solution can indeed be expressed in the desired form.