Domain of ln(e^x-1)

Question

I need to find the domain of ln(e^x-1)

Answer 1

To find the domain of ln(e^x-1) we should look at the different parts of this expression one at a time, since this function of x is really a composition of functions. We know the domain of e^x is (-∞,+∞) since we can put a number to any power. The range of e^x is only positive numbers (0,+∞). Now lets consider the domain and range of e^x-1. The domain will remain the same since we can raise e to any power and subtract one. The range is now (-1,+∞).

Now if we consider ln(e^x-1), we know that we cannot take the natural log of anything less that or equal to zero (since we cannot raise e to any power to get 0 or less.) So the domain of this function begins after

e^x-1=0 ⇒ e^x=1 ⇒ x=0. We can take the natural log of anything larger than 0, so any larger value works ∴

our domain is (0,+∞).

Answer 2

The domain is the possible values that x can have. You can only do ln of a positive number so the e^x - 1 must be positive. Se if you can find what values of x make e^x - 1 positive. That will be your answer.