Raising e to the natural log of a function in intdeterminate forms

Question

Why would one need to raise e to the power of a function that is in indeterminate form? I understand the concept of logarithmic differentiation (taking the ln of both sides of the equation), but I am unfamiliar with why it is legal to change the equation to e raised to the power of the natural log of the function. When should this method be used, and what is its advantage over simply taking the ln of both sides?

Answer 1

Here are two common mathematical strategies:

1. Sometimes you want to do the same thing to both sides of an equation to get another equation. To do this, simply apply a function to both sides of the equation.
2. Other times, you want to "apply an identity", to change just one side of the equation without altering the other side.

Although these are clearly very similar strategies, nonetheless they have their different uses. You should learn to use both.

So, for example, the very very very useful identity that is at the heart of your question is $$x = e^{\ln(x)}, \quad x > 0 $$ and you can see how this identity is used in the example of @S.Koch.

Answer 2

This is useful because you still have the same function. For example, this is the common method to differentiate $x^x $ because: $$x^x=e^{\log x^x}=e^{x\log x} $$ You can differentiate the right side using chain and product rule of derivatives, but you can't differentiate the left side directly.

As for why it is allowed: simply a consequence of definition: When $a^b=c $ we define $\log_a c := b $. Substituting $b $ in the former formula we get $$c=a^{\log_a c} $$