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Thread: Solving functions equation

  1. #1
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    Solving functions equation

    Need help with the following.

    Solving functions equation-screen-shot-2013-05-12-8.38.05-pm.png

    Thanks
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Solving functions equation

    a) $\displaystyle 25^{x-3}=5^x$

    You can get the same base here since $\displaystyle 25=5^2$, so that you may write:

    $\displaystyle (5^2)^{x-3}=5^x$

    $\displaystyle 5^{2(x-3)}=5^x$

    Now equate exponents, and solve for $\displaystyle x$.

    b) $\displaystyle 2\left(\frac{1}{2} \right)^x-\left(\frac{1}{4} \right)^x=1$

    Use the substitution $\displaystyle u=\left(\frac{1}{2} \right)^x$ so that you may nor write:

    $\displaystyle 2u-u^2=1$

    Now, solve the resulting quadratic, then back substitute for $\displaystyle u$.

    c) $\displaystyle 3\ln(2x+5)=9$

    Divide through by 3 to get:

    $\displaystyle \ln(2x+5)=3$

    Now convert from logarithmic to exponential form and solve for $\displaystyle x$.

    d) $\displaystyle e^{(x-2)^2}=2$

    Convert from exponential to logarithmic form:

    $\displaystyle (x-2)^2=\ln(2)$

    Now solve for $\displaystyle x$ using the square root property.

    e) $\displaystyle 6^{-7x}=5$

    Take the natural log of both sides to get:

    $\displaystyle -7x\ln(6)=\ln(5)$

    Now solve for $\displaystyle x$.
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