inverse functions (ln x and e^x)

[Math wiz ya rite 09]

Recall that y = ln(x) and y = e^x are inverse functions. THe graph of y = ln(x) looks as though it might be approaching a horizontal asymptote. Write an argument based on the graph of y = e^x to explain why it does not.


My responce:
Since they are inverse functions, every (x,y) pair of ln(x) corresponds to a pair (y,x) of e^x. If ln(x) approaches a horizontal asymptote, that means that y has a limit. That means there is a limit on the x values of f(x). This is ridiculous, so obviously there is no horizontal asymptote for ln(x).


Does any other way to answer the question. Maybe somehting more mathematical than mine?

Thanks

 

[pka]

I think that your attempt will work with some additional clarifications. You should mention that x is approaching infinity. And that means that y is bounded by the asymptotic value. What does that mean for \(\displaystyle e^x\)?

Use what you started. Here is some mathematics that may help.
If ln(x) approaches a horizontal asymptote, then \(\displaystyle \left( {\exists B > 0} \right)\left( {\forall x} \right)\left[ {\ln (x) < B} \right]\).

But that means \(\displaystyle e^{\ln (x)} < e^B \quad \Rightarrow \quad x < e^B \mbox{ as } x \to \infty\).