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Math Help - a^x = b^x + c

  1. #1
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    a^x = b^x + c

    how does one solve an equation of the type a^x = b^x + c??

    for the life of me i cant do it.
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  2. #2
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    Quote Originally Posted by ayoyoayoyo View Post
    how does one solve an equation of the type a^x = b^x + c??

    for the life of me i cant do it.
    To get you started, take the natural log of both sides of the equation.
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  3. #3
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    tried that but the constant term got in the way
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  4. #4
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    Quote Originally Posted by ayoyoayoyo View Post
    tried that but the constant term got in the way
    Isn't it true that ln (ax) = ln(a) + ln(x)?
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  5. #5
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    i can not see how that pertains to the question at hand

    we have ln(b^x+c)
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  6. #6
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    Quote Originally Posted by wonderboy1953 View Post
    Isn't it true that ln (ax) = ln(a) + ln(x)?
    I'll go further:

    ln(a^x) = ln(b^x +c)= ln(b^x)ln(c), then

    xln(a) = xln(b)ln(c); ln(c) = xln(a)/xln(b) = ln(a)/ln(b) = ln(a) - ln(b)
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  7. #7
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    The logarithm of a sum is not something you can simplify. The identity only works the other way.

    With this problem, I'd probably go for a numerical solution using Newton-Raphson. Of course, this assumes you know a, b, and c, and that a and b are both positive. Is that the case?
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  8. #8
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    so no analytic solution?
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  9. #9
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    None of which I am aware. Both Mathematica and WolframAlpha fail to give an analytical solution. I should point out that not every combination of a, b, and c will admit a solution, even if all of them are positive.
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  10. #10
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    how do you use newton's to find the solution? isnt newtons only used to find the roots of a f()?
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  11. #11
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    Right, but you can always convert an equation-solving problem into a root-finding problem by throwing everything on to one side of the equation thus:

    f(x)=a^{x}-b^{x}-c=0.

    Then you use Newton-Raphson on f(x).

    If you have convergence problems, and your solutions are oscillating wildly, then you probably don't have a solution for that particular combination of a, b, and c. That's a little warning sign you can look for.
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