Dear Uncle Colin,

I need to solve $5^{2x+2} +16\cdot 15^x - 9^{x+1} = 0$ but I’ve hit a dead end! Can you help?

- Puzzling Out Wild Exponent Relations

Hi, POWER, and thanks for your message!

From the working you sent, it looks like you’ve picked a good strategy: you’ve let $X = 5^x$ and $Y = 3^x$ to make things a bit tidier, turning the whole thing into a quadratic in two variables.

That gives you $25 X^2 + 16 XY - 9Y^2 = 0$, which looks like a mess.

Fortunately, it factorises as $(25X - 9Y)(X + Y)=0$, so either $\frac{Y}{X} = \frac{25}{9}$ or $X = -Y$.

The last of those is impossible (exponentials are always positive), so it must be the case that $\frac{3^x}{5^x} = \frac{25}{9}$

There are several ways to tackle this; by inspection, $x=-2$ works. Alternatively, you can take logs to say $x\ln \left(\frac{3}{5}\right) = \ln\left(\frac{25}{9}\right)$ and let your calculator do the not-exactly-heavy lifting.

Hope that helps!

- Uncle Colin