Verified answer

I think what you intended to type was that Log(z) = ln(2) - (π/2)i. So, using the exponential function

exp(Log(z)) = z = exp(ln(2) - (π/2)i) = exp(ln(2))exp(-(π/2)i) = 2 e^[(π/2)i].

So, z = 2 e^[(π/2)i].

The key to this problem is to remember the motivation for the definition of the log function for complex numbers:

z = |z| e^(i arg z) --> log z = log [|z| e^(i arg z)] = log |z| + i arg z = ln |z| + i arg z (In order for this to work out, you must remember that you are using log base e, not base 10 as you are accustomed to for this notation. This is why we switched to ln above: because log base e of a real number is just the ln of that real number). If you are using the principle argument of z, Arg z, then

Log z = ln |z| + i Arg z, and this is single valued (whereas log z is multi-valued).

Just raise both sides to the power 10:

z = 10^{ln(2)} - π/2i}