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Math Help - Need help with Logarithm and Inverse Function

  1. #1
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    Need help with Logarithm and Inverse Function

    I am having difficulties understanding how to solve the following problems:

    1) Express as a single logarithm and, if possible, simplify:

    \frac{3}{2}ln4x^6 - \frac{4}{5}ln2y^{10}

    The book shows the answer as: ln\frac{2^{11/5}x^9}{y^8}

    I understand how the x^6 turns into x^9, but how does the 4 in \frac{3}{2}ln4x^6 turn into 2^{11/5}?

    2) Express as a single logarithm:

    7lnx + 3(lny^2 - lnz^3)

    Since #1 is similar to #2, I am unsure on how to correctly solve this problem.

    3) Find the inverse function of:

    f(x) = \frac{3x}{x-3}

    When I do the math, it shows that the inverse function is exactly the same as the original function? How is that possible?

    I would appreciate it greatly if you could answer the questions above step-by-step.

    Thanks
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  2. #2
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    Quote Originally Posted by Ineedhelpwithlogs View Post
    I am having difficulties understanding how to solve the following problems:

    1) Express as a single logarithm and, if possible, simplify:

    \frac{3}{2}ln4x^6 - \frac{4}{5}ln2y^{10}

    The book shows the answer as: ln\frac{2^{11/5}x^9}{y^8}

    I understand how the x^6 turns into x^9, but how does the 4 in \frac{3}{2}ln4x^6 turn into 2^{11/5}?
    \frac{3}{2}\ln(4x^6) - \frac{4}{5}\ln(2y^{10})

    \ln(4x^6)^{\frac{3}{2}} - \ln(2y^{10})^{\frac{4}{5}}

    \ln\left(\frac{(4x^6)^{\frac{3}{2}}}{(2y^{10})^{\f  rac{4}{5}}}\right)

    Now simplify
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  3. #3
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    Quote Originally Posted by Ineedhelpwithlogs View Post
    3) Find the inverse function of:

    f(x) = \frac{3x}{x-3}

    When I do the math, it shows that the inverse function is exactly the same as the original function? How is that possible?

    I would appreciate it greatly if you could answer the questions above step-by-step.

    Thanks
    y = f(x) = \frac{3x}{x-3}

    swap x and y and then solve for y

    x  = \frac{3y}{y-3}

    (y-3)x  = 3y

    yx-3x  = 3y

    -3x  = 3y-yx

    -3x  = y(3-x)

    Can you solve for y?
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  4. #4
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    Quote Originally Posted by pickslides View Post
    \frac{3}{2}\ln(4x^6) - \frac{4}{5}\ln(2y^{10})

    \ln(4x^6)^{\frac{3}{2}} - \ln(2y^{10})^{\frac{4}{5}}

    \ln\left(\frac{(4x^6)^{\frac{3}{2}}}{(2y^{10})^{\f  rac{4}{5}}}\right)

    Now simplify
    Simplifying is where I get lost - sorry, I should have mentioned that.
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  5. #5
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    Quote Originally Posted by pickslides View Post
    y = f(x) = \frac{3x}{x-3}

    swap x and y and then solve for y

    x  = \frac{3y}{y-3}

    (y-3)x  = 3y

    yx-3x  = 3y

    -3x  = 3y-yx

    -3x  = y(3-x)

    Can you solve for y?
    y = \frac{-3x}{3 - x}
    f^{-1}(x) = \frac{-3x}{3 - x}

    Thank you, pickslides. I see where I made my mistake.
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  6. #6
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    \frac{3}{2}\ln(4x^6) - \frac{4}{5}\ln(2y^{10})

    \ln(4x^6)^{\frac{3}{2}} - \ln(2y^{10})^{\frac{4}{5}}

    \ln\left(\frac{(4x^6)^{\frac{3}{2}}}{(2y^{10})^{\f  rac{4}{5}}}\right)

    \ln\left(\frac{4^{\frac{3}{2}}x^{6\times \frac{3}{2}}}{2^{\frac{4}{5}}y^{10\times\frac{4}{5  }}}\right)
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  7. #7
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    Quote Originally Posted by pickslides View Post
    \frac{3}{2}\ln(4x^6) - \frac{4}{5}\ln(2y^{10})

    \ln(4x^6)^{\frac{3}{2}} - \ln(2y^{10})^{\frac{4}{5}}

    \ln\left(\frac{(4x^6)^{\frac{3}{2}}}{(2y^{10})^{\f  rac{4}{5}}}\right)

    \ln\left(\frac{4^{\frac{3}{2}}x^{6\times \frac{3}{2}}}{2^{\frac{4}{5}}y^{10\times\frac{4}{5  }}}\right)

    \ln\left(\frac{4^{\frac{3}{2}}x^{9}}{2^{\frac{4}{5  }}y^{8}}\right)

    then divide 4 by 2 to get:

    \ln\left(\frac{2^{\frac{3}{2}}x^{9}}{{\frac{4}{5}}  y^{8}}\right)

    Now, I don't know what to do next (and I'm not sure if that last step was correct).
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