Question

Inverse logarithmic function

Answer 1

So they want the inverse of \[F(x)=\ln \sqrt{x}\]
So the inverse is
\[x=\ln \sqrt{y}\]

Answer 2

yep,
now isolate y

Answer 3

\(\Large a=\ln c \implies c = e^a \)

Answer 4

\[y=\left( e ^{x} \right)^{2}\] the 2 can go down, right?

Answer 5

y=2e^x

Answer 6

not actually

Answer 7

\(\Large (e^x)^2 = e^{2x}\)

Answer 8

oops, another silly mistake

Answer 9

Can someone help me demostrate that (FoFinverse)(x)=(FinverseoF)(x)=x ?

Answer 10

demonstrate ?
you have a function for verifying that ?

Answer 11

The one we just did

Answer 12

\[\ln \sqrt{e ^{2x}} = e ^{2\ln \sqrt{x}}=x\]

Answer 13

so, f inverse = e^2x

just plug in x =ln \sqrt x

oh, you did it :)

Answer 14

But how and why? (I think that was a good statement)

Answer 15

And how would that be x?

Answer 16

ln x^m = m ln x

Answer 17

ln e =1

Answer 18

\(\sqrt {(e^x)^2} = |e^x| \\ \ln |e^x| = x \ln e = x\)

Answer 19

ok ?

Answer 20

so the left one is kinda obvious, so it is x

Answer 21

Oh...and the other one is with
\[a ^{logaN}=\]

Answer 22

N

Answer 23

\(\huge a^{\log_aN}=N\)

Answer 24

yes

Answer 25

:) Now I understand, thanks for fifth time, just one more question :P

Answer 26

welcome ^_^

Answer 27

Which would be the domain and the range of the first, normal function?

Answer 28

domain of ln sqrt x ?

Answer 29

since its sqrt x, x>= 0
since its ln, sqrt x >0
in all, x>0

Answer 30

so the range of the inverse is x>0

Answer 31

Wish me good luck tomorrow

Answer 32

yes!
best of luck! hope you get full marks :)

Answer 33

I hope so, this will be a hard examn...all types of functions, trigonometric identities, operations with functions

Answer 34

if you have practices enough, nothing is hard :)

Answer 35

Thanks for the help all night!

Answer 36

you're welcome ^_^