complex analysis - 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?

Answer

We can try converting the formula to base $10$.

Let $\log$ be the natural (i.e., base $e$) logarithm. Then $10 = e^{\log10}$, so:

$$
10^{ix} = e^{ix\log10}=\cos(x\log10) + i\sin(x\log10)
$$