Real root of exponential equation

CAT 2019 Slot 2 · Quantitative Ability · Medium · Logarithms

The real root of the equation $2^{6x} + 2^{3x+2} - 21 = 0$ is

Answer

Let $y = 2^{3x}$ . Then the equation becomes: $y^2 + 4y - 21 = 0$ $(y + 7)(y - 3) = 0$

Since $2^{3x} > 0$ for all real $x$ , $y$ must be positive. Thus $y = 3$ . $2^{3x} = 3 \implies 3x = \log_2 3 \implies x = \frac{\log_2 3}{3}$

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