Problem

Solve the equation \(2^{\cos x} = 8\) for \(x\), given that \(0 \leq x \leq 2\pi\).

Answer

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Answer

Step 4: Since the range of the cosine function is \([-1, 1]\), there is no solution for this equation.

Steps

Step 1 ：Step 1: Take the natural logarithm on both sides: \(\ln(2^{\cos x}) = \ln 8\).

Step 2 ：Step 2: Use the property of logarithms to bring the exponent to the front: \(\cos x \cdot \ln 2 = \ln 8\).

Step 3 ：Step 3: Note that \(\ln 8 = 3 \ln 2\), so the equation simplifies to \(\cos x = 3\).

Step 4 ：Step 4: Since the range of the cosine function is \([-1, 1]\), there is no solution for this equation.

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