# Why does a^x = e^(x(lna))

by jessepye
Tags: exlna
 P: 418 Why does a^x = e^(x(lna)) EDIT You got it before I typed this I think this is right, I'm just trying to remember it off the top of my head as my textbook is in school. Let the value of $a^{x}$ be equal to $y$ $a^{x} = y$ Take natural log of both sides $ln(a^{x}) = ln(y)$ Then we can bring the exponent out of the bracket $x * ln(a) = ln(y)$ Then we put both sides as the power of e to cancel the ln on the right $e^{x * ln(a)} = e^{ln(y)}$ $e^{x * ln(a)} = y$ Then since $a^{x} = y$ we sub that in for y and get $e^{x * ln(a)} = a^{x}$
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,568 Another way to see the same thing is to note that $aln(x)= ln(x^a)$ so that $e^{xln(a)}= e^{ln(a^x)}$. Then, because "$f(x)= e^x$" and "$g(x)= ln(x)$" are inverse functions, $e^{ln(a^x)}= a^x$.