Horribly Confused With Complex Logarithms

[MrBillyShears]

I'm getting myself all confused with complex logarithms. I'll try to explain why. One identity with complex logarithms is ln(z^c)=cln(z)+2πik, with k an integer. This is, of course, a more general case of ln(e^c)=c+2πik, but it doesn't always work the same! Let's say we are evaluating ln(e^i). Using the latter identity, it is i+2πik, which is, logically, the correct answer, but using the first identity, you get iln(e)+2πik, which is i(1+2πin)+2πik=i+2πn+2πik...! What! Obviously e^(i+2π) doesn't equal e^i. Another example, ln(1)=ln(e^2πi)=2πi(1+2πin)+2πik=2πi+4π^2n+2 πik

And, I have another problem. I have this when I try to solve an equation 10^z=e^πi, so I take ln of both sides zln(10)=πi+2πik and then z=(πi+2πik)/ln(10), where ln(10) in the denominator is infinite answered and will give solutions that don't work! I'm clearly doing something wrong, so someone please help me!

 

[mfb]

One identity with complex logarithms is ln(z^c)=cln(z)+2πik, with k an integer.

If this identity allows complex c (check that!), then you have to be more careful which branch of the logarithm you choose on the right side.

where ln(10) in the denominator is infinite answered and will give solutions that don't work!

A multi-valued log is not the exact inverse function of 10^z for reasonable definitions of 10^z. If you use the same freedom to define 10^z as exp(ln(10)z) with a multi-valued ln(10) then they should work.