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osanseviero

  • 2 years ago

Inverse logarithmic function

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  1. osanseviero
    • 2 years ago
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    So they want the inverse of \[F(x)=\ln \sqrt{x}\] So the inverse is \[x=\ln \sqrt{y}\]

  2. hartnn
    • 2 years ago
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    yep, now isolate y

  3. hartnn
    • 2 years ago
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    \(\Large a=\ln c \implies c = e^a \)

  4. osanseviero
    • 2 years ago
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    \[y=\left( e ^{x} \right)^{2}\] the 2 can go down, right?

  5. osanseviero
    • 2 years ago
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    y=2e^x

  6. hartnn
    • 2 years ago
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    not actually

  7. hartnn
    • 2 years ago
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    \(\Large (e^x)^2 = e^{2x}\)

  8. osanseviero
    • 2 years ago
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    oops, another silly mistake

  9. osanseviero
    • 2 years ago
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    Can someone help me demostrate that (FoFinverse)(x)=(FinverseoF)(x)=x ?

  10. hartnn
    • 2 years ago
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    demonstrate ? you have a function for verifying that ?

  11. osanseviero
    • 2 years ago
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    The one we just did

  12. osanseviero
    • 2 years ago
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    \[\ln \sqrt{e ^{2x}} = e ^{2\ln \sqrt{x}}=x\]

  13. hartnn
    • 2 years ago
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    so, f inverse = e^2x just plug in x =ln \sqrt x oh, you did it :)

  14. osanseviero
    • 2 years ago
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    But how and why? (I think that was a good statement)

  15. osanseviero
    • 2 years ago
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    And how would that be x?

  16. hartnn
    • 2 years ago
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    ln x^m = m ln x

  17. hartnn
    • 2 years ago
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    ln e =1

  18. hartnn
    • 2 years ago
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    \(\sqrt {(e^x)^2} = |e^x| \\ \ln |e^x| = x \ln e = x\)

  19. hartnn
    • 2 years ago
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    ok ?

  20. osanseviero
    • 2 years ago
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    so the left one is kinda obvious, so it is x

  21. osanseviero
    • 2 years ago
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    Oh...and the other one is with \[a ^{logaN}=\]

  22. osanseviero
    • 2 years ago
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    N

  23. hartnn
    • 2 years ago
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    \(\huge a^{\log_aN}=N\)

  24. hartnn
    • 2 years ago
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    yes

  25. osanseviero
    • 2 years ago
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    :) Now I understand, thanks for fifth time, just one more question :P

  26. hartnn
    • 2 years ago
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    welcome ^_^

  27. osanseviero
    • 2 years ago
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    Which would be the domain and the range of the first, normal function?

  28. hartnn
    • 2 years ago
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    domain of ln sqrt x ?

  29. hartnn
    • 2 years ago
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    since its sqrt x, x>= 0 since its ln, sqrt x >0 in all, x>0

  30. osanseviero
    • 2 years ago
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    so the range of the inverse is x>0

  31. osanseviero
    • 2 years ago
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    Wish me good luck tomorrow

  32. hartnn
    • 2 years ago
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    yes! best of luck! hope you get full marks :)

  33. osanseviero
    • 2 years ago
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    I hope so, this will be a hard examn...all types of functions, trigonometric identities, operations with functions

  34. hartnn
    • 2 years ago
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    if you have practices enough, nothing is hard :)

  35. osanseviero
    • 2 years ago
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    Thanks for the help all night!

  36. hartnn
    • 2 years ago
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    you're welcome ^_^

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