exponential functions

[katyt]

this might seem like a stupid question, but I cannot remember for the life of me the rules for this. ok so i'm solving a differential equation and i end up with:
s=e^(.5ln(2t)+t^2+C)
I'm having trouble remembering the rules for exponential functions. I can't figure out how to rewrite/simplify this. Any help?
Thanks! Katy

 

[soroban]

Hello, Katy!

That was not a stupid question . . . This is a tricky one.

i'm solving a differential equation and i end up with: .$\displaystyle s\:=\:e^{[0.5\ln(2t)\,+\,t^2\,+\,c]}$

I can't figure out how to rewrite/simplify this.

Recall that: .$\displaystyle e^{a+b}\:=\:e^a\,\cdot\,e^b$

So: .$\displaystyle e^{[0.5\ln(2t)\,+\,t^2\,+\,c]}\;=\;e^{0.5\ln(2t)}\,\cdot\,e^{t^2}\,\cdot\,e^c$

Since $\displaystyle e^c$ is just another constant $\displaystyle C$, we have: .$\displaystyle C\cdot e^{t^2}\cdot e^{0.5\ln(2t)}$

Then: .$\displaystyle C\cdot e^{t^2}\cdot e^{\ln(2t)^{\frac{1}{2}}}\:=\:C\cdot e^{t^2}\cdot(2t)^{\frac{1}{2}}\:=\:C\cdot e^{t^2}\cdot\sqrt{2t}$