How to simplify expressions in Mathematica with log and exp in them?

[Master1022]

TL;DR

How can I get the expression $log(exp(z))$ to return $z$ in Mathematica?

Hi,

This is a pretty simple question, but I am new to Mathematica so I am not sure if I am missing something obvious.

Question: How do I make the expression $e^{log(z)}$ return z?

Attempt:
I have used all of the following combinations and all of them return $e^{log(z)}$. Are there any tips people have? I have seen some people online use some elaborate functions, but I am just looking for a simple(r) solution.

[CODE title="Mathematica"]e^log[z]
E^log[z]
PowerExpand[E^log[z]]
Simplify[PowerExpand[E^log[z]]][/CODE]

Thanks in advance.

 

[jedishrfu]

Maybe solve() will work:

https://reference.wolfram.com/language/ref/Solve.html

 

[Master1022]

Maybe solve() will work:

https://reference.wolfram.com/language/ref/Solve.html

I'll give that a try. Thanks!

 

[renormalize]

PowerExpand[] works for me. Be sure to capitalize Log , Exp and E !

In[1]:= PowerExpand[Log[Exp[z]]]

Out[1]= z

 

[Master1022]

PowerExpand[] works for me. Be sure to capitalize Log , Exp and E !

In[1]:= PowerExpand[Log[Exp[z]]]

Out[1]= z

Oh wow, that does work! Many thanks. Do you know of any reason why using an upper case Log[] inside a Solve[] function could lead to an error, but then I changed it to log[] (lower case) and the error went away).

 

[renormalize]

Do you know of any reason why using an upper case Log[] inside a Solve[] function could lead to an error, but then I changed it to log[] (lower case) and the error went away).

To answer I'd have to see how you set up your Solve[] statement. Can you share it?

 

[Mark44]

@Master1022, in your first post you asked two different questions:
How can I get the expression $\log(exp(z))$ to return z in Mathematica?
Question: How do I make the expression $e^{\log(z)} return z? First off, is z a complex number? A variable named z is often used to represent a complex number, so if you use z to represent a real number, that can lead to confusion amongst readers. Mathematically,$\log(exp(z))$is always defined and is equal to z, but the opposite order in the composition may not be defined due to$\log(z)## not being defined. If z is real, log(z) is defined only for z > 0.

If z is complex, the Mathematica documentation says this:

Log[z] has a branch cut discontinuity in the complex z plane running from $-\infty$ to 0.