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Math Help - Logs- finding inverses

  1. #1
    Member smmmc's Avatar
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    Logs- finding inverses

    Help!

    how do i do these questions.

    Determine the invers of these:
    a)y=3log10(x)
    b)y=2log10(3x)


    Thanks
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  2. #2
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    Quote Originally Posted by smmmc View Post
    Help!

    how do i do these questions.

    Determine the invers of these:
    a)y=3log10(x)
    b)y=2log10(3x)


    Thanks
    Hi

    \log_{10} x = \frac{y}{3} \rightarrow x = 10^{\frac{y}{3}}
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  3. #3
    Member smmmc's Avatar
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    how about y=10^3x ?


    thanks
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  4. #4
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    Quote Originally Posted by smmmc View Post
    how about y=10^3x ?


    thanks
    You original function was f(x)= 3 log_{10}(x). The basic definition of "inverse functions" requires that f^{-1}(f(x))= x and f(f^{-1}(x))= x. 3 log_{10}(10^{3x})= 3(3x)= 9x, not x. Similarly 10^{3 (3 log_{10}(x)}= 10^{9log_{10}(x)}= 10^{log_{10}(x^9)}= x^9, not x.

    A standard method of finding the inverse of a function given as y= f(x) is
    1) Swap x and y (that's the key point).
    2) Solve the equation for y.
    3) If there is a unique solution for y as a function of x, that is the inverse.

    Here, y= 3 log_{10}(x). Swapping x and y, x= 3log_{10}(y). To solve for y, first, divide both sides by 3: \frac{x}{3}= log_{10}(x). Now take 10 to the power of both sides: 10^{\frac{x}{3}}= 10^{log_{10}(y)}= y. Since y= 10^{\frac{x}{10}}, that is the inverse function.
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