1. ## Complex logarithm question

The question:
Find the Cartesian form of $Log((e^{\frac{3\pi i}{4}})^2)$ (note that this is the principle branch logarithm)

My attempt:
I'm not sure how to deal with the square. I recall that the index and log laws do not hold in this context. So how do I deal with it?

2. Originally Posted by Glitch
The question:
Find the Cartesian form of $Log((e^{\frac{3\pi i}{4}})^2)$ (note that this is the principle branch logarithm)

My attempt:
I'm not sure how to deal with the square. I recall that the index and log laws do not hold in this context. So how do I deal with it?
Isn't it true that $\displaystyle e^{\left(\frac{3\pi i}{4}}\right)^2 = e^{\frac{3 \pi i}{2}}$ ....

3. Ahh yes, now that I double checked, the index law does hold, but the log laws don't. I'm think I'm confusing principle value exponents with exponentials. :/

I continued and did this:

$Log(e^{\frac{3\pi i}{2}})$
= $ln|1| + (\frac{3}{2} + 2k)\pi i : k \in Z$
However it's the principle log, so k = 0.

= $\frac{3\pi}{2}i$

However, the solution is $\frac{-i\pi}{2}$

Any suggestions?

4. Originally Posted by Glitch
Ahh yes, now that I double checked, the index law does hold, but the log laws don't. I'm think I'm confusing principle value exponents with exponentials. :/

I continued and did this:

$Log(e^{\frac{3\pi i}{2}})$
= $ln|1| + (\frac{3}{2} + 2k)\pi i : k \in Z$
However it's the principle log, so k = 0.

= $\frac{3\pi}{2}i$

However, the solution is $\frac{-i\pi}{2}$

Any suggestions?
Well, the principle log would require using the principle argument, would it not? Look up what definition of principle argument you're meant to be using.

5. I think I understand now. Thank you.