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Power series
- Thread starter Trumbone
- Start date
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...
\(\displaystyle a^x = e^{x \ln(a)} = \sum_{n=0}^{\infty} \frac{(x\ln a)^n}{n!}\)
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...
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 \(\displaystyle \ln(a^x) = x \ln(a)\)
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 \(\displaystyle \ln(a^x) = x \ln(a)\)