# Where do I go wrong in solving this mathematical expression?

**URL:** https://discourse.julialang.org/t/where-do-i-go-wrong-in-solving-this-mathematical-expression/92266
**Category:** Offtopic
**Tags:** math
**Created:** [December 29, 2022, 10:18am UTC](https://discourse.julialang.org/t/where-do-i-go-wrong-in-solving-this-mathematical-expression/92266 "2022-12-29T10:18:11Z")
**Posts on this page:** 3
**Page:** 1

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### Author: ![Ahmed\_Salih](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ahmed_salih/32/206579_2.png) [@Ahmed\_Salih](https://discourse.julialang.org/u/Ahmed_Salih)
#### Post date: [December 29, 2022, 10:18am UTC](https://discourse.julialang.org/t/where-do-i-go-wrong-in-solving-this-mathematical-expression/92266/1 "2022-12-29T10:18:11Z")

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Hello!

I was not sure if I should have chosen “Off-topic” or this “Mathematical” section, so please move post to appropriate section.

For now it is not a programming question I have, but more about how to solve this equation for `ρ`:

![image](https://global.discourse-cdn.com/julialang/original/3X/4/c/4cd790c370e197c1d1fdb18ef96618eb86491b01.png)

I’ve done the steps I am sure of, to get:

![image](https://global.discourse-cdn.com/julialang/original/3X/d/3/d380e6ac6a8e50a80a054933d68d3bbd3536ad51.png)

Then I know I would have to use some logarithm to get rid of the exponent:

![image](https://global.discourse-cdn.com/julialang/original/3X/3/2/32dc866140b568489212b31f07f6c9b392bb6534.png)

Then I found a log rule stating that ln(x/y) = ln(x) - ln(y):

![image](https://global.discourse-cdn.com/julialang/original/3X/c/e/ceed3c24a8d9dd5603f69a3b6fe9044f5aaaaaea.png)  
 ![image](https://global.discourse-cdn.com/julialang/original/3X/9/3/931a94b865980082ef46417e0ed0aa362d12a6c5.png)

And to cancel the log I re-raise to an exponent and get the following expression:

![image](https://global.discourse-cdn.com/julialang/original/3X/5/8/58509295f79cc555e139f7f016faefcff6c9f796.png)

It seems though that I have made a mistake somwhere since wolfram alpha returns:

 ![image](https://global.discourse-cdn.com/julialang/original/3X/2/f/2f32a4cb99b750ab4b03fda4a9d124728575536b.png)

Where ρ\_0 is multiplied on, instead of added too.

If someone would be kind enough to explain to me where my mistake lies, I would really appreciate it 🙂

Kind regards

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### Author: ![cvanaret](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/cvanaret/32/11594_2.png) [@cvanaret](https://discourse.julialang.org/u/cvanaret)
#### Post date: [December 29, 2022, 10:30am UTC](https://discourse.julialang.org/t/where-do-i-go-wrong-in-solving-this-mathematical-expression/92266/2 "2022-12-29T10:30:33Z")

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e^{a+b} = e^a e^b 😉

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### Author: ![Ahmed\_Salih](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ahmed_salih/32/206579_2.png) [@Ahmed\_Salih](https://discourse.julialang.org/u/Ahmed_Salih)
#### Post date: [December 29, 2022, 10:34am UTC](https://discourse.julialang.org/t/where-do-i-go-wrong-in-solving-this-mathematical-expression/92266/3 "2022-12-29T10:34:47Z")

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Ah okay, got it!

I should have raised the whole RHS to exp, and not the terms individually. So this is wrong, it should have been the equation below:

![image](https://global.discourse-cdn.com/julialang/original/3X/b/a/ba787e9828095039af45a35b51219ef74d0dcf71.png)

And then the equation you wrote holds and I get the right answer 🙂

Thank you!