# Euler's formula with base 10?

Does Euler's formula ($e^{ix} = \cos x + i \sin x$) work in base $10$? If it does, how could I express it?

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• One may always write $$10^{ix}=e^{ix\ln(10)}=\cos[ix\ln(10)]+i\sin[ix\ln(10)].$$ – Olivier Oloa Jan 2 '18 at 14:26
• @mdewey What ?? – Rebellos Jan 2 '18 at 14:40
• If you are ok, you can accept the answer and set as solved. Thanks! – gimusi Jan 3 '18 at 17:29

We can try converting the formula to base $10$.
Let $\log$ be the natural (i.e., base $e$) logarithm. Then $10 = e^{\log10}$, so:
$$10^{ix} = e^{ix\log10}=\cos(x\log10) + i\sin(x\log10)$$
$$10^{ix}=(e^{\log 10})^{ix} =(e^{ix})^{\log 10} = (\cos x + i \sin x)^{\log 10}$$