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

 

[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]

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).

 

[CellCoree]

, while sin(x) has a range of -1 to 1. This is not the case with ln(x) and e^x, so why does the same logic not apply?

I can explain this using mathematical principles and properties.

Firstly, it is important to understand that ln(x) and e^x are inverse functions, meaning that they "undo" each other. This is similar to how multiplication and division are inverse operations. For example, 2 x 3 = 6 and 6 ÷ 2 = 3.

Now, let's look at the equation ln(e^x) = x. This is true because ln(x) and e^x are inverse functions of each other. When we take the natural logarithm of e^x, we are essentially "undoing" the exponential function, and we are left with just x. This is why the equation is true.

On the other hand, e^ln(x) = x may seem confusing at first. However, we can understand this by looking at the properties of logarithms and exponentials. The property we need to focus on here is the power rule, which states that when we have an exponential expression raised to another exponent, we can multiply the exponents. In this case, we have e^ln(x), which can be rewritten as (e^1)^ln(x). Using the power rule, we can rewrite this as e^(1 * ln(x)), which simplifies to e^ln(x). And as we know, e^ln(x) is equal to x.

In summary, the equations ln(e^x) = x and e^ln(x) = x both make sense when we understand the properties of logarithms and exponentials. The key difference is that in the first equation, we are "undoing" the exponential function, while in the second equation, we are using the power rule to simplify the expression.