How does e^(2ln(t)) equal t^2?

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The discussion centers on the confusion regarding the equation e^(2ln|t|) and its equivalence to t^2. The key point is the application of logarithmic properties, specifically that a ln b = ln(b^a) and e^(ln a) = a. By correctly applying these rules, e^(2ln|t|) simplifies to e^(ln(t^2)), which equals t^2. The misunderstanding arose from incorrectly manipulating the exponential and logarithmic functions. Clarifying these properties resolved the confusion and confirmed the equality.
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ok guys, i don't see how e^2ln|t| = t^2 can someone explain it to me please? Seems so easy but i don't see it.
 
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ok, so far i tried this. I know e^(lnx) = x


so, i broke the e^(2ln|t|) into to parts:


e^2 times e^(ln|t|) which equals t (from the top identity)


so I am left with e^2 times t. which is te^2

but the book says it equals t^2..so what happened to the e? (exponential function)
 
CinderBlockFist said:
ok, so far i tried this. I know e^(lnx) = x


so, i broke the e^(2ln|t|) into to parts:


e^2 times e^(ln|t|) which equals t (from the top identity)


so I am left with e^2 times t. which is te^2

but the book says it equals t^2..so what happened to the e? (exponential function)

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.
 
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First look at this rule for logarithms:

\log_b(a^x)=x\log_b(a).

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

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

Those two rules together will give you the answer.
 
SWEEET! THANK YOU GUYS. I got my laws of exponents mixed up.
 
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