## osanseviero 2 years ago Inverse logarithmic function

1. osanseviero

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

2. hartnn

yep, now isolate y

3. hartnn

$$\Large a=\ln c \implies c = e^a$$

4. osanseviero

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

5. osanseviero

y=2e^x

6. hartnn

not actually

7. hartnn

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

8. osanseviero

oops, another silly mistake

9. osanseviero

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

10. hartnn

demonstrate ? you have a function for verifying that ?

11. osanseviero

The one we just did

12. osanseviero

$\ln \sqrt{e ^{2x}} = e ^{2\ln \sqrt{x}}=x$

13. hartnn

so, f inverse = e^2x just plug in x =ln \sqrt x oh, you did it :)

14. osanseviero

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

15. osanseviero

And how would that be x?

16. hartnn

ln x^m = m ln x

17. hartnn

ln e =1

18. hartnn

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

19. hartnn

ok ?

20. osanseviero

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

21. osanseviero

Oh...and the other one is with $a ^{logaN}=$

22. osanseviero

N

23. hartnn

$$\huge a^{\log_aN}=N$$

24. hartnn

yes

25. osanseviero

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

26. hartnn

welcome ^_^

27. osanseviero

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

28. hartnn

domain of ln sqrt x ?

29. hartnn

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

30. osanseviero

so the range of the inverse is x>0

31. osanseviero

Wish me good luck tomorrow

32. hartnn

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

33. osanseviero

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

34. hartnn

if you have practices enough, nothing is hard :)

35. osanseviero

Thanks for the help all night!

36. hartnn

you're welcome ^_^