5

Solve for x:

$$16^x+36^x=81^x$$

This question was given by one of the student in the lesson. Can this question be solved?

A possible approach is to take the log

$$x\log(81)=\log(16^x+36^x)$$

It is impossible to isolate the $x$ on the RHS.

N. F. Taussig
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Sibawayh
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    Related : https://math.stackexchange.com/questions/384090/find-all-real-numbers-x-for-which-frac8x27x12x18x-frac76 https://math.stackexchange.com/questions/1356236/what-is-the-solution-to-the-equation-9x-6x-2-cdot-4x-0 https://math.stackexchange.com/questions/1237837/solve-left-sqrt34-sqrt15-rightx-left-sqrt34-sqrt15-right https://math.stackexchange.com/questions/1330422/how-to-solve-this-equation-for-x-left-sqrt2-sqrt3-rightx-left-sq https://math.stackexchange.com/questions/1778037/solve-for-x-9x-6x-4x1-2-hints-only-please – lab bhattacharjee Dec 26 '19 at 06:04

1 Answers1

16

We can rewrite it as

$$\left(\frac{16}{81}\right)^x+\left(\frac{36}{81}\right)^x=1 \implies\left(\frac{4}{9}\right)^{2x}+\left(\frac{4}{9}\right)^x-1=0$$

Let $y=\left(\frac{4}{9}\right)^x$.

Solve the quadratic in $y$, then substitute $x = \frac{\ln y}{\ln(\frac49)}$

user26857
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19aksh
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