log10i

Log10 i refers to the logarithm base 10 of the complex number i, where i is the imaginary unit with i^2 = -1. In complex analysis, logarithms are inherently multi-valued because of the periodicity of the complex exponential. The relation log10 z = log z / log 10 holds, where log denotes the complex logarithm (the natural logarithm in the numerator) and log 10 is the real logarithm of 10.

For a general complex number z, log z = ln|z| + i Arg(z) + 2kπi, with k an arbitrary integer and Arg(z) the argument of z. Consequently, log10 z = [ln|z| + i Arg(z) + 2kπi] / ln 10. When z = i, its magnitude |i| = 1, so ln|i| = 0, and Arg(i) = π/2 + 2πk. Therefore log10(i) = i (π/2 + 2πk) / ln 10 for any integer k.

The principal value (taking the principal value of the argument Arg in (−π, π]) is obtained with k = 0, giving log10(i) = i (π/2) / ln 10 ≈ 0.682188i. In general, the values are purely imaginary, with real part zero and imaginary part (π/2 + 2πk) / ln 10.

This reflects the broader property that the complex logarithm is multi-valued; in practical contexts, the principal value is used unless a multi-valued solution is required.