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How do you simplify:

5*e^(-10x) - 3*e^(-20x) = 2

I'm not sure if I can take natural log of both sides to bring down the exponents.

  • Calculus -

    I will assume you want to "solve" for x

    let e^(-10x) = y , then e^(-20x) = (e^(-10x)^2 = y^2
    then you have

    5y - 3y^2 = 2
    3y^2 - 5y + 2 = 0
    (y-1)(3y-2) = 0
    y = 1 or y = 2/3

    so e^(-10x) = 1 or e^(-10x) = 2/3

    if e^(-10x) = 1
    ln e^(-10x) = ln 1 = 0
    -10x lne = 0
    -10x = 0
    x = 0

    if e^(-10x) = 2/3
    -10x = ln2 - ln3
    x = -(l2 - ln3)/10 = appr .04055

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