Power series

[Trumbone]

Does anyone know how to find the power series of a[sup:1y7eze82]x[/sup:1y7eze82]?
and/or
Show that e[sup:1y7eze82]s+t[/sup:1y7eze82]= e[sup:1y7eze82]s[/sup:1y7eze82]e[sup:1y7eze82]t[/sup:1y7eze82]using algebra with power series?

 

[daon]

For the first one:

$\displaystyle a^x = e^{x \ln(a)} = \sum_{n=0}^{\infty} \frac{(x\ln a)^n}{n!}$

For the second, are you supposed to use binomial expansion? It's probably possible that way, though I've personally only seen this derived from the limit definition of exp...

 

[Trumbone]

Thank you so much for your help, but I am a little confused about how you went from a[sup:gdjw5b1n]x[/sup:gdjw5b1n] to e[sup:gdjw5b1n]xln(a)[/sup:gdjw5b1n].

 

[daon]

For any 1-1 function f on a member k of its domain, $\displaystyle f^{-1}(f(k)) = k$

Since $\displaystyle e^x$ and $\displaystyle \ln(x)$ are inverse functions and $\displaystyle a>0$ means we have $\displaystyle e^{\ln(a^x)} = a^x$.

We also know $\displaystyle \ln(a^x) = x \ln(a)$