Question about why ln(e^x) =/= x

physicsernaw

Why is ln(e^x) =/= x?

The domain and range of the LHS are the same as the RHS, so I don't understand why this equation is false, where e^ln(x) = x, and the LHS and RHS of this does not have the same domain...

I know that e^x and ln(x) are inverse functions, so please don't only tell me this. Why does e^ln(x) = x, while ln(e^x) =/= x?

EDIT:
Like, I understand why sin^-1(sin(x)) =/= x, whereas sin(sin^-1(x)) = x, and this is because sin^-1(x) has a range of -pi/2 to pi/2

Last edited:

Vorde

They are equal.

ln(e^x) = x*ln(e) = x

What/who told you differently?

physicsernaw

They are equal.

ln(e^x) = x*ln(e) = x

What/who told you differently?
I could have sworn wolfram alpha was telling me the equation is false, but I just tried again and it told me the equation is indeed true Mark44

Mentor
It's the other way around that you run into domain considerations:

$$e^{\ln x} = x$$

The left side is defined only for x > 0 (considering only the real-valued log function).

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving