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

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

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