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I was given a question on homework that I'm somewhat confused about.

$7^x = 9^{x+7}$

The teacher said that there was no solution, which can be shown as the lines don't ever intersect. However it looks like the lines would intersect at $-\infty$, and I am aware that infinity becomes important in limits. Would this be a valid solution, or am I getting ahead of myself?

(This is a highschool math class)

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  • $\begingroup$ Speaking about "solutions" of such an equation, $\pm\infty$ are not admitted. And the two lines don't intersect, indeed ("intersect at infinity" is just a way of speaking). $\endgroup$
    – Anne Bauval
    Dec 9, 2022 at 16:09
  • $\begingroup$ Is $x$ supposed to be an integer? $\endgroup$
    – John Douma
    Dec 9, 2022 at 16:17

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Taking logarithms, $$x\log7 = (x+7)\log9,$$ from which $$x = \frac{7\log 9}{\log 7 - \log 9} \approx -61.2.$$ Maybe your teacher means there is no positive solution, or that there is no whole-number solution.

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