Real root of exponential equation

CAT 2019 Slot 2 · QA · Medium · Logarithms

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The real root of the equation 26x+23x+2−21=02^{6x} + 2^{3x+2} - 21 = 0 is

  1. A.

    log⁡233\frac{\log_2 3}{3}

  2. B.

    log⁡29\log_2 9

  3. C.

    log⁡273\frac{\log_2 7}{3}

  4. D.

    log⁡227\log_2 27

Answer

A

Explanation

Let y=23xy = 2^{3x}. Then the equation becomes: y2+4y−21=0y^2 + 4y - 21 = 0 (y+7)(y−3)=0(y + 7)(y - 3) = 0

Since 23x>02^{3x} > 0 for all real xx, yy must be positive. Thus y=3y = 3. 23x=3  ⟹  3x=log⁡23  ⟹  x=log⁡2332^{3x} = 3 \implies 3x = \log_2 3 \implies x = \frac{\log_2 3}{3}

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