Dear Uncle Colin,

I need to solve $3^{x+2} + 45\br{6^x} - 9\br{2^x}=0$ and I’m completely stuck. How would you tackle it?

Producing A Sketch Could Accelerate Learning

Hi, PASCAL, and thanks for your message!

I think this is a question best approached by simplifying a little at a time.

My first step would be to look at $3^{x+2}$ and say “that’s the same as $9\br{3^x}$” – if I replace that, then all of the powers are $x$s and that’s probably a good thing. It also makes all of the coefficients a multiple of 9, which is also good.

I get $9\br{3^x} + 45\br{6^x} - 9\br{2^x} = 0$ – and dividing out the 9 gives $3^x + 5\br{6^x} - 2^x = 0$.

Divide through by $6^x$: $2^{-x} + 5 - 3^{-x} = 0$, or $3^{-x} - 2^{-x} = 5$.

A solution of $x = -2$ almost jumps out here. Is it the only one?

I can’t immediately tell. Let’s mentally sketch $y = 3^{-x} - 2^{-x}$.

  • Domain? It’s defined for all $x$.
  • Axes? It passes through the origin, moving downwards.
  • Turning points? Let’s come back to that.
  • Asymptotes? When $x$ is large and positive, it approaches zero from below. Moving backwards into negative $x$, we rise without limit.
  • Shape? I think it comes down from large $y$ when $x$ is negative, passes through the origin on its way down, then rises slowly back to approach zero from below.

We can find the turning point if we’re really interested:

  • $3^{-x}\ln(3) = 2^{-x}\ln(2)$
  • $\br{\frac{3}{2}}^x = \frac{\ln(3)}{\ln(2)}$
  • $x = \frac{\ln\br{\frac{\ln(3)}{\ln(2)}}}{\ln(\frac{3}{2})} \approx 1.136$

Ugh. But reassuringly, there’s a turning point right where we wanted one.

In any case: the graph is monotonically decreasing to zero for negative $x$, and negative for positive $x$, so the only place it can possibly equal 5 is at $x = -2$.

Hope that helps!

- Uncle Colin