# Thread: Solving |e^z| using complex numbers

1. ## Solving |e^z| using complex numbers

Solve $\displaystyle |e^z| = 2$ for z

Attempt:

$\displaystyle \sqrt{e^{2z}} = e^z$

Therefore:

$\displaystyle e^z = 2e^{0 + i2\pi k}$
$\displaystyle = e^{ln2 + i2\pi k}$

$\displaystyle z = ln2 +i2\pi k, k E Z$

but the correct answer was $\displaystyle z = ln2 + ik$

2. You need to notice that $\displaystyle \left| {e^z } \right| = \left| {e^{x + yi} } \right| = e^x \left| {\cos (y) + i\sin (y)} \right| = e^x$

3. Originally Posted by Plato
You need to notice that $\displaystyle \left| {e^z } \right| = \left| {e^{x + yi} } \right| = e^x \left| {\cos (y) + i\sin (y)} \right| = e^x$
But then how can you now solve for z like the orignal question asked to, since you dont have a z variable anymore?

Using what you told me I got:

$\displaystyle e^x$
$\displaystyle = 2$
$\displaystyle = e^{ln2}e^{0 + 2\pi k i}$

$\displaystyle x = ln2 + 2\pi k i, k E Z$

but my answer obviously isnt right.

4. Originally Posted by SyNtHeSiS
But then how can you now solve for z like the orignal question asked to, since you dont have a z variable anymore?

Using what you told me I got:

$\displaystyle e^x$
$\displaystyle = 2$
$\displaystyle = e^{ln2}e^{0 + 2\pi k i}$

$\displaystyle x = ln2 + 2\pi k i, k E Z$

but my answer obviously isnt right.
You're expected to realise that x is real ....

5. Oh I see, but why is it that you add $\displaystyle iy$ to the solution and not $\displaystyle 2 \pi k i$?

6. Originally Posted by SyNtHeSiS
Oh I see, but why is it that you add $\displaystyle iy$ to the solution and not $\displaystyle 2 \pi k i$?
No, see post #2.
The answer is $\displaystyle z=\ln(2)+yi$ for any $\displaystyle y\in \mathbb{R}.$

7. Originally Posted by Plato
No, see post #2.
The answer is $\displaystyle z=\ln(2)+yi$ for any $\displaystyle y\in \mathbb{R}.$
I understand what you did but can you just tell me why this method is wrong:

$\displaystyle 2 = 2(cos0 + isin0)$
$\displaystyle = 2e^0$

I read somewhere in my notes that you always add $\displaystyle 2\pi k i$to $\displaystyle \theta$ and in this case my $\displaystyle \theta$ is 0, so then I did the following:
$\displaystyle = 2e^{0 + 2\pi k i}$

$\displaystyle e^x = e^{ln2}e^{2\pi k i}$

$\displaystyle x = ln2 + 2\pi k i, k E R$

I can see that k must be R since x is a real number.

But I just want to know why you dont add $\displaystyle 2\pi ki$ here

8. Originally Posted by SyNtHeSiS
I read somewhere in my notes that you always add $\displaystyle 2\pi k i$to $\displaystyle \theta$ and in this case my $\displaystyle \theta$ is 0, so then I did the following: $\displaystyle = 2e^{0 + 2\pi k i}$
$\displaystyle e^x = e^{ln2}e^{2\pi k i}$
$\displaystyle x = ln2 + 2\pi k i, k E R$
I can see that k must be R since x is a real number.
But I just want to know why you dont add $\displaystyle 2\pi ki$ here
I don't know what your notes say, of course.
But you are missing the whole idea here.

$\displaystyle \left|e^{\ln(2)+yi}\right|=\left|e^{\ln(2)}\right| \left|\cos(y)+i\sin(y)\right|=2$ for all $\displaystyle y\in \mathbb{R}$.