Solve the exponential equation for x 3^7x+4=(1/27)^x-3

Question

Solve the exponential equation for x
3^7x+4=(1/27)^x-3

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Khoii Minh 5 years 2021-09-03T10:55:06+00:00 1 Answers 17 views 0

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  1. Vodka
    0
    2021-09-03T10:56:42+00:00
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    Answer:

    The solution is x = \frac{1}{2}

    Step-by-step explanation:

    To solve this exponential equation, we must write both sides as powers of 3.

    The equation given is:

    3^{7x + 4} = (\frac{1}{27})^{x-3}

    Since 3^3 = 27, we have that 3^{-3} = \frac{1}{3^3} = \frac{1}{27}, so\frac{1}{27} = 3^{-3}

    Then

    3^{7x + 4} = (\frac{1}{27})^{x-3}

    3^{7x + 4} = (3^{-3})^{x-3}

    3^{7x + 4} = 3^{-3(x-3)}

    3^{7x + 4} = 3^{-3x + 9}

    Since both sides are now powers of 3, we can equal them:

    7x + 4 = -3x + 9

    10x = 5

    x = \frac{5}{10}

    x = \frac{1}{2}

    The solution is x = \frac{1}{2}

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