 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/1/solution

t = 2^{x} > 0

4^{x} = (2^{x})^{2}

reduces the problem to the quadratic equation

t^{2} - t - 2 = (t - 2) (t + 1) = 0.

Positivity excludes

t = - 1

, so

2^{x} = 2

2^{x}

is strictly increasing, the unique answer is

x = 1

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