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Thread: Exponential Equations

  1. #1
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    Exponential Equations

    Solve each equation below. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places.

    (1) (cuberoot{2})^(2 - x) = 2^(x^2)

    (2) e^(x + 3) = pi^(x)
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  2. #2
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    $\displaystyle (\sqrt[3]{2})^{2-x} = 2^{x^2}$

    $\displaystyle (\sqrt[3]{2})^{2-x} = [(\sqrt[3]{2})^3]^{x^2}$

    $\displaystyle (\sqrt[3]{2})^{2-x} = (\sqrt[3]{2})^{3x^2}$

    so ... what can you say about $\displaystyle 2-x$ and $\displaystyle 3x^2$ ?


    $\displaystyle e^{x+3} = \pi^x$

    $\displaystyle \ln(e^{x+3}) = \ln(\pi^x)$

    $\displaystyle x+3 = x\ln{\pi}$

    $\displaystyle 3 = x\ln{\pi} - x$

    $\displaystyle 3 = x(\ln{\pi} - 1)$

    $\displaystyle \frac{3}{\ln{\pi} - 1} = x$
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  3. #3
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    Much better....

    This is much better than the other reply.
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