E^(2lnt) = t^2 Whyyy?

you have

e^2 e^(ln t) = e^(2+ln t),

which is incorrect.


you want to use the properties:


a ln b = ln b^a

and

e^ln a = a.


the rest should be straightfoward.

 

First look at this rule for logarithms:

[itex]\log_b(a^x)=x\log_b(a)[/itex].

Now apply that to your exponent:[itex]2\ln |t|[/itex]. What do you get?

Then note that [itex]f(x)=\ln (x)[/itex] and [itex]g(x)=e^x[/itex] are inverse functions, which means that [itex]f(g(x))=g(f(x))=x[/itex].

Those two rules together will give you the answer.