The number of solutions of the equation 3^2^2x+3^2^2x=81, 0 x4 is:

Question

The number of solutions of the equation $3^{2	an^2x}+3^{2\sec^2x}=81$, $0\le x\le\dfrac\pi4$ is:

Accepted Answer

Since $\sec^2x=1+\tan^2x$: $3^{2\tan^2x}+3^{3}\cdot3^{2\tan^2x}=81 \implies 28\cdot3^{2\tan^2x}=81 \implies 3^{2\tan^2x}=\tfrac{81}{28}$

$$\tan^2x = \frac{\ln(81/28)}{2\ln3} \implies \tan x = \sqrt{\frac{\ln(81/28)}{2\ln3}} \in (0,1)$$

Since this value lies in $(0,1)$, there is exactly one solution in $\left[0,\dfrac\pi4\right]$.

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