Can Euler still help?

Algebra Level 4

If the expression i π + e + 1 i\pi + e + 1 can be expressed in the form ln ⁡ ( k ) \ln(k) where k k is a real number, determine k k to three decimal places.


The answer is -41.194.

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Efren Medallo by Efren Medallo , 26, Philippines

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1 solution

ln ⁡ ( k ) = π i + e + 1 k = e π i + e + 1 k = e π i e e + 1 \ln(k)=\pi i+e+1 \\ k=e^{\pi i+e+1} \\ k=e^{\pi i}e^{e+1}

By Euler's formula, e i x = cos ⁡ x + i sin ⁡ x e^{ix}=\cos x+i\sin x , so e π i = cos ⁡ π + i sin ⁡ π = − 1 e^{\pi i}=\cos \pi + i \sin \pi=-1 , and our expression becomes:

k = − e e + 1 k ≈ − 41.194 k=-e^{e+1} \\ k \approx \boxed{-41.194}

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