Solving unknown involving powers.

Question

For the question,

$5^x = 5^{x+2} - 5^3$. I was asked to solve for $x$.

I reached, $5^3 = 5^{x+2} - 5^x$ unable to progress further.

Answer 1

$5^{x+2}-5^{x}=5^{3}\iff$

$5^{x}\cdot5^{2}-5^{x}=5^{3}\iff$

$5^{x}\cdot(5^{2}-1)=5^{3}\iff$

$5^{x}=\frac{5^{3}}{5^{2}-1}\iff$

$5^{x}=\frac{125}{24}\iff$

$x=\log_5\frac{125}{24}\iff$

$x=\log_5125-\log_524\iff$

$x=3-\log_524$

Answer 2

We have that $$5^3 = 5^{x+2} - 5^x=5^x\cdot 5^2-5^x=25\cdot 5^x-5^x=(25-1)\cdot 5^x=24\cdot 5^x \implies 5^x=\frac{5^3}{24}.$$ Taking the logarithm in base 5, we finally get $$x=\log_5\left(\frac{5^3}{24}\right)=\log_5(5^3)-\log_5(24)=3-\log_5(24).$$