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Math Help - Basic exponents

  1. #1
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    Basic exponents

    3^x + 5^x = 12

    I have taken logs on both sides, used the power rule to get

    xln3 + x ln5 = ln12

    factored out x on the LHS to give x(ln3 + ln5) = ln12

    combined logs

    x(ln15) = ln12

    divided through by ln15

    x = (ln12)/(ln15)

    The answer I get doesnt match what I want. Where have I gone wrong please?
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by 200001 View Post
    3^x \color{red}+ 5^x = 12
    I fear that this equation cannot be solved symbolically.


    I have taken logs on both sides, used the power rule to get

    xln3 + x ln5 = ln12
    That would be correct, if the original equation had been 3^x\cdot 5^x=12.

    factored out x on the LHS to give x(ln3 + ln5) = ln12

    combined logs

    x(ln15) = ln12

    divided through by ln15

    x = (ln12)/(ln15)
    That's correct, provided, as I wrote above, the original equation really was 3^x\cdot 5^x=12.

    The answer I get doesnt match what I want. Where have I gone wrong please?
    Either you have gone wrong in the very first step when you assumed that \ln(3^x+5^x)=x\ln 3+x\ln 5, or you have gone wrong in believing that the solution that you have found is not correct (or, rather, that it is "not what you want").
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  3. #3
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    Thanks for your reply.
    I run it through wolfram and their answer was the same as the given answer.

    (3^x)+(5^x)&#x3 d;12 - Wolfram|Alpha

    How would I approach this?
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  4. #4
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    Quote Originally Posted by 200001 View Post
    Thanks for your reply.
    I run it through wolfram and their answer was the same as the given answer.

    (3^x)+(5^x)&#x3 d;12 - Wolfram|Alpha

    How would I approach this?
    Use the solution in Wolframalpha as a hint: "Numerical solution: ..."

    Use Newton's method to get an approximate value of x:

    1. Change the equation to: f(x)=3^x+5^x-12 and solve the equation f(x) = 0 for x.

    2. Newton's method:

    x_{n+1}=x_n - \frac{f(x_n)}{f'(x_n)}

    I assume that you know how to differentiate an exponential function(?).

    3. Use x_0 = 1. After 4 steps of iteration you'll have the result given by Wolframalpha.
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  5. #5
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    Sorry I am not as mathematically developed to appreciate a numerical solution as a hint.
    Thank you though for your subsequent direction
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