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Inverse logarithmic function

Mathematics
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So they want the inverse of \[F(x)=\ln \sqrt{x}\] So the inverse is \[x=\ln \sqrt{y}\]
yep, now isolate y
\(\Large a=\ln c \implies c = e^a \)

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Other answers:

\[y=\left( e ^{x} \right)^{2}\] the 2 can go down, right?
y=2e^x
not actually
\(\Large (e^x)^2 = e^{2x}\)
oops, another silly mistake
Can someone help me demostrate that (FoFinverse)(x)=(FinverseoF)(x)=x ?
demonstrate ? you have a function for verifying that ?
The one we just did
\[\ln \sqrt{e ^{2x}} = e ^{2\ln \sqrt{x}}=x\]
so, f inverse = e^2x just plug in x =ln \sqrt x oh, you did it :)
But how and why? (I think that was a good statement)
And how would that be x?
ln x^m = m ln x
ln e =1
\(\sqrt {(e^x)^2} = |e^x| \\ \ln |e^x| = x \ln e = x\)
ok ?
so the left one is kinda obvious, so it is x
Oh...and the other one is with \[a ^{logaN}=\]
N
\(\huge a^{\log_aN}=N\)
yes
:) Now I understand, thanks for fifth time, just one more question :P
welcome ^_^
Which would be the domain and the range of the first, normal function?
domain of ln sqrt x ?
since its sqrt x, x>= 0 since its ln, sqrt x >0 in all, x>0
so the range of the inverse is x>0
Wish me good luck tomorrow
yes! best of luck! hope you get full marks :)
I hope so, this will be a hard examn...all types of functions, trigonometric identities, operations with functions
if you have practices enough, nothing is hard :)
Thanks for the help all night!
you're welcome ^_^

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