J JPJ New member Joined Jan 24, 2012 Messages 10 Feb 13, 2012 #1 Why can I say that \(\displaystyle 2^(ln x) = x^(ln 2)\) ? ps.: Sorry about the latex typos, couldn't get it to work...meant it like 2^(ln x) = x^(ln 2).
Why can I say that \(\displaystyle 2^(ln x) = x^(ln 2)\) ? ps.: Sorry about the latex typos, couldn't get it to work...meant it like 2^(ln x) = x^(ln 2).
J JPJ New member Joined Jan 24, 2012 Messages 10 Feb 13, 2012 #2 Nevermind about it...just figured it out: 2^lnx = (e^Ln2)^Lnx = e^(Ln2*Lnx) = (e^Lnx)^Ln2 = x^Ln2
S soroban Elite Member Joined Jan 28, 2005 Messages 5,584 Feb 13, 2012 #3 Hello, JPJ! \(\displaystyle \text{Why can I say: }\:2^{\ln x} \:=\:x^{\ln 2}\,?\) Click to expand... Never mind about it . . . just figured it out: . . \(\displaystyle 2^{\ln x} \:=\: \left(e^{\ln2}\right)^{\ln x} \:=\:e^{\ln2\cdot\ln x} \:=\: \left(e^{\ln x}\right)^{\ln 2} \:=\: x^{\ln 2}\) Click to expand... Elegant proof . . . Good work!
Hello, JPJ! \(\displaystyle \text{Why can I say: }\:2^{\ln x} \:=\:x^{\ln 2}\,?\) Click to expand... Never mind about it . . . just figured it out: . . \(\displaystyle 2^{\ln x} \:=\: \left(e^{\ln2}\right)^{\ln x} \:=\:e^{\ln2\cdot\ln x} \:=\: \left(e^{\ln x}\right)^{\ln 2} \:=\: x^{\ln 2}\) Click to expand... Elegant proof . . . Good work!
