How do you use the graphing calculator to solve 2^(-x-3)=3^(x+1)?

Apr 29, 2018

$x \approx - 1.774 \ldots$

Explanation:

We can plot both functions on the same plot and see where they intersect.

Socratic doesn't have a good way to plot two functions as far as I know, so you can see a good plot here.

We could also do this analytically using logarithms:
$\ln \left({2}^{- x - 3}\right) = \ln \left({3}^{x + 1}\right)$
$\ln \left(2\right) \left(- x - 3\right) = \ln \left(3\right) \left(x + 1\right)$
$\ln \left(3\right) \left(x + 1\right) + \ln \left(2\right) \left(x + 3\right) = 0$
$\ln \left(6\right) x + \ln \left(24\right) = 0$
$x = - \frac{\ln \left(24\right)}{\ln \left(6\right)} = - {\log}_{6} \left(24\right) \approx - 1.774 \ldots$

which is exactly where we saw it in the above.