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Math Help - How to solve for x??: (2^x) + (3^x) = 13

  1. #1
    Member aldrincabrera's Avatar
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    How to solve for x??: (2^x) + (3^x) = 13

    How to solve for x??:

    (2^x) + (3^x) = 13
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    Re: How to solve for x??: (2^x) + (3^x) = 13

    2^2= ?, 3^2= ?.
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    Re: How to solve for x??: (2^x) + (3^x) = 13

    Quote Originally Posted by HallsofIvy View Post
    2^2= ?, 3^2= ?.
    It is clear that x=2 but how do we arrive at this conclusion using algebraic method?
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    Re: How to solve for x??: (2^x) + (3^x) = 13

    Quote Originally Posted by MatthiasChampagne View Post
    It is clear that x=2 but how do we arrive at this conclusion using algebraic method?
    You don't arrive at this conclusion using algebraic methods.
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  5. #5
    Member aldrincabrera's Avatar
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    Re: How to solve for x??: (2^x) + (3^x) = 13

    Can't we use logarithm here sirs??
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    Re: How to solve for x??: (2^x) + (3^x) = 13

    Hi,
    no, use logaritnm don't helps.
    The problem haven't regulary algebric solution,

    but simply x=1 or x=2 or x=3?
    Probe, and see...
    Last edited by StefanTM; October 8th 2012 at 06:22 AM.
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  7. #7
    Member aldrincabrera's Avatar
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    Re: How to solve for x??: (2^x) + (3^x) = 13

    thanks sirs...so basically,it's just trial and error???
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    MHF Contributor MarkFL's Avatar
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    Re: How to solve for x??: (2^x) + (3^x) = 13

    Knowing there will only be one root from the properties of exponential functions, we could use a root finding technique, such as Newton's method, which converges to x=2 in 10 iterations on my calculator with an initial guess of x_0=0.

    I would use inspection first to see if a rational root exists, but say the equation had instead been:

    2^x+3^x=12

    Now, using Newton's method, we find:

    x\approx1.91768594489
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    Member aldrincabrera's Avatar
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    Re: How to solve for x??: (2^x) + (3^x) = 13

    ,.,thank you for that idea sir,.but i just got a little confused in using Newton's method,.,can you please elaborate a little on the steps sir??thnx
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    MHF Contributor MarkFL's Avatar
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    Re: How to solve for x??: (2^x) + (3^x) = 13

    Have you studied differential calculus? Newton's method uses differentiation in its recursive approach. Newton's method essentially finds the x-intercept of the line tangent to the function for which we are trying to find a root at a value of x and uses that as a better approximation. Then this method is repeated recursively to converge to a root of the function. There are functions for which this doesn't always work for particular values of x, but for well-behaved functions like exponential functions, this is a good technique to use.
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  11. #11
    Member aldrincabrera's Avatar
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    Re: How to solve for x??: (2^x) + (3^x) = 13

    ,.yes sir,.but when i substituted the values,.i got it all wrong.,.,i have the derivative (ln(3)*(3^x)) + (ln(2)*(2^x)) for the function,.but i got messed up for the substitution. help
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    Re: How to solve for x??: (2^x) + (3^x) = 13

    Quote Originally Posted by aldrincabrera View Post
    ,.yes sir,.but when i substituted the values,.i got it all wrong.,.,i have the derivative (ln(3)*(3^x)) + (ln(2)*(2^x)) for the function,.but i got messed up for the substitution. help
    If you know calculus, then looking at the derivative tells you that 2^t+3^t-13 in everywhere increasing. It means that t=2 is the only root.

    So no need for Newton's method.

    BTW Why was this question posted in basic algebra forum?
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  13. #13
    MHF Contributor MarkFL's Avatar
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    Re: How to solve for x??: (2^x) + (3^x) = 13

    Quote Originally Posted by aldrincabrera View Post
    ,.yes sir,.but when i substituted the values,.i got it all wrong.,.,i have the derivative (ln(3)*(3^x)) + (ln(2)*(2^x)) for the function,.but i got messed up for the substitution. help
    You have the correct derivative.

    For the function:

    f(x)=2^x+3^x-k=0

    f'(x)=\ln(2)2^x+\ln(3)3^x and so by Newton's method, we have the recursion:

    x_{n+1}=x_n-\frac{2^{x_n}+3^{x_n}-k}{\ln(2)2^{x_n}+\ln(3)3^{x_n}}

    As stated, when k=13 there certainly is no need for this technique, but I simply wanted to give a method for approximating the root when it is irrational, since you seemed to be interested in an "algebraic" method.
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    MHF Contributor MarkFL's Avatar
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    Re: How to solve for x??: (2^x) + (3^x) = 13

    You obviously didn't read the previous posts AND you have replaced the tacky link in your signature that you were asked to remove...nicely done!
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