Question 1009730: A WebWork questions has asked if the following functions have inverses:
1. ln(x^18)
2. 18ln(x)
As far as I know, they're both the same because the first can be re-written as the second according to the log exponent property. However, WW claims that the first one does not have an inverse, while the second one does.
I know that 18 is even, so ln(x^18) is defined for x<0 (where ln(x) isn't defined) and has two parts that are symmetric about the y-axis (so the inverse function wouldn't pass the vertical line test).
That said, If ln isn't defined for x<0, how can there even be a function for ln(x^18)? Furthermore, isn't the second function (f(x)=18ln(x)) just an alternate form? If I rewrote 18ln(x) to ln(x^18), wouldn't it become undefined as well?
Or is it that when we start off with ln(x^18), it's implicitly assumed that the value of x can be either negative or positive because a positive exponent means the result will be the same? In 18ln(x), however, we've started with an exponent of 1, which is odd, so x values will respect their signs.
But wouldn't this cause some trouble during algebraic manipulation, though? Would we just have to respect the original form? I mean, if I had a situation where I had to isolate x, I would technically be allowed to change ln(x^18) to 18ln(x) and vice versa, right?
I would really appreciate a clarification on the difference between the two functions.
Answer by addingup(3677)   (Show Source):
You can put this solution on YOUR website! I'm with you, these are two ways of writing the same equation:ln(x^18) or 18ln(x). Again, there may be some dark secret I'm missing, but to me the inverse for either one is the same: e^x/18
J
|
|
|
