Where is the mistake?

I know that log function is defined only for positive reals but please help me understand where i go wrong in the following proof : TO PROVE : $\ln\big(i) = \frac{i\pi}{2}$ PROOF : $\ln(i) = \ln(e^{\frac{i\pi}{2}})$ but we know that $\ln(e^x) = x$ and hence $\ln(i) = \ln(e^{\frac{i\pi}{2}}) = \frac{i\pi}{2}$ Hence proved . Is it correct ? Although , i am sure it is wrong but which step is precisely not allowed here?

#Algebra

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

it's not the same the exponentation or logharythmic functions with the complex numbers, that the exponentation or logharytmic functions with real numbers. Look at this : $i = e^{ \frac {i \pi}{2}} =e^{ \frac {5i \pi}{2}}$ ...

Guillermo Templado - 4 years, 5 months ago

thanks for the help :)

Ujjwal Mani Tripathi - 4 years, 5 months ago

If you need something about me, you can ask to me, and if I can help you, I'll help you.

Guillermo Templado - 4 years, 5 months ago

@Guillermo Templado – thank you ...

Ujjwal Mani Tripathi - 4 years, 5 months ago

@Ujjwal Mani Tripathi – thank you for you,of course

Guillermo Templado - 4 years, 5 months ago

@Guillermo Templado – this discussion has turned out to be fun......all are becoming friends :)

Ujjwal Mani Tripathi - 4 years, 5 months ago

@Ujjwal Mani Tripathi – thank you again friend... I like that we are all friends

Guillermo Templado - 4 years, 5 months ago

Hello my friend again

Guillermo Templado - 4 years, 5 months ago

What makes you think that $\ln i \neq \frac{ i \pi } { 2}$ ?

Also, be very careful with complex exponentiation and logarithms.

Calvin Lin Staff - 4 years, 5 months ago

@Guillermo Templado sir , this doubt arose from your solution to imaginary^imaginary . Can you please help me?

Ujjwal Mani Tripathi - 4 years, 5 months ago

Yes, I'm going on there...

Guillermo Templado - 4 years, 5 months ago

The sequence of steps that you took essentially work, but you have to be clear if you are working with "equality of multi-valued functions" or "evaluate at primitive branch".

To be clear, the multi-valued function $\ln i$ is indeed equal to $\frac{ i \pi }{2} + 2k \pi i$ .

Calvin Lin Staff - 4 years, 5 months ago

Calvin, please, I want to be your friend, but you don't make me easy it... $\frac{i \pi}{2} + 2k\pi \neq i\cdot(\frac{\pi}{2} + 2k \pi)$

Guillermo Templado - 4 years, 5 months ago

Ah yes, fixed the typo. thanks.

Calvin Lin Staff - 4 years, 5 months ago

@Calvin Lin – Don't worry, can we be friends?

Guillermo Templado - 4 years, 5 months ago

@Guillermo Templado – Of course!

Calvin Lin Staff - 4 years, 5 months ago

@Calvin Lin – Thank you very much.

Guillermo Templado - 4 years, 5 months ago

@Guillermo Templado – Calvin, I would write great wikis, but I have some problems:

a) Where can I start?

b) I can't do everything, I'm not God/Godness or...

c) For example, suppose I want to write a wiki on numerical analysis, before doing this, would I have to define what a vectorial space is, would I have to give a topology or abstract algebra course first, would I have to establish some prior knowledge of mathematical analysis?...

d) In short, what would I have to do?

Guillermo Templado - 4 years, 5 months ago

@Guillermo Templado –

Here is a list of wikis that we're working on. If there are any that you're interested in helping out with, ping #wikis.
If you're working on a new wiki and need help with developing the outline, let me know.
Don't worry about it. Focus on what you're good at / what you're interested in.
Do not worry about ensuring that foundational material exists when developing out a page. We can fill that in subsequently. and use "red links" to indicate that wiki pages are undeveloped. E.g. In the Gram Schmidt wiki, we would assume knowledge of vector space, and link out to it.
Where do you want to start?

Calvin Lin Staff - 4 years, 5 months ago

@Calvin Lin – Good answer, let's start at the beginning, there is already a story about this, a teacher from Tai-chi taught it to me. Suppose I want to talk about arithmetic theory: where do I have to start? Defining Peano's axioms?... Now suppose I want to talk about set theory, would I have to give a theory about what a class is?

P.S.- Imagine that I am nobody, neither a mathematician nor a physicist ..

Guillermo Templado - 4 years, 5 months ago

@Guillermo Templado – Look, we can make one thing, I like so much topology, and here I don't see Topology. What about if I start talking about Topology, with more or less a notion of a "set", I'll "definite it", I don't want Russell's paradox again... do you remember what happened...? and later I continue talking about Topology, and when I need help, I can email you or stop... Remember, we are free if we respect ourselves... what do you think about this?... I have other question: where is Mateo Matija...? I can't get see him... It's important for me...

Guillermo Templado - 4 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$