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Math Help - Simple equation help?

  1. #1
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    Simple equation help?

    Hi there,

    This popped up in my exam the other day, absolutely no idea how to solve it using secondary school knowledge:

    2^x = -x
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  2. #2
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    Re: Simple equation help?

    You will need to solve it numerically, guess and check or technology.

    Graph y = 2^x & y= -x , to get a rough idea on where they intersect.
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  3. #3
    MHF Contributor chisigma's Avatar
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    Re: Simple equation help?

    Quote Originally Posted by eskimogenius View Post
    Hi there,

    This popped up in my exam the other day, absolutely no idea how to solve it using secondary school knowledge:

    2^x = -x
    The equation You have written is trascendental and its solution has to be found numerically. A non well popular but elegant way to solve it is to writre the equation as f(x)= -2^{x}-x=0 [th sign '-' has a precise scope...] and then compute for n large enough the n-th term of the sequence defined by the 'recursive relation' ...

    x_{n+1}=-2^{x_{n}}\ ;\ x_{0}=0 (1)

    In few step You arrive at the solution x \sim -.641186. The reason for which x_{n} tends to the [real] solution of Your equation will be explained in a tutorial post written in the section 'Discrete mathematics'...

    Kind regards

    \chi \sigma
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  4. #4
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    Re: Simple equation help?

    Quote Originally Posted by chisigma View Post
    The equation You have written is trascendental and its solution has to be found numerically. A non well popular but elegant way to solve it is to writre the equation as f(x)= -2^{x}-x=0 [th sign '-' has a precise scope...] and then compute for n large enough the n-th term of the sequence defined by the 'recursive relation' ...

    x_{n+1}=-2^{x_{n}}\ ;\ x_{0}=0 (1)

    In few step You arrive at the solution x \sim -.641186. The reason for which x_{n} tends to the [real] solution of Your equation will be explained in a tutorial post written in the section 'Discrete mathematics'...

    Kind regards

    \chi \sigma
    Sorry, I have no idea what this means, or how to do this.

    I have graphed the lines, and realised that they intersect between 0 and -1, however, I don't know how to arrive at the solution.

    Can it be done using simple log laws?
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  5. #5
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    Re: Simple equation help?

    the solution is probably transcendental. Do you know newton's method?
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  6. #6
    MHF Contributor chisigma's Avatar
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    Re: Simple equation help?

    Quote Originally Posted by eskimogenius View Post
    Sorry, I have no idea what this means, or how to do this.

    I have graphed the lines, and realised that they intersect between 0 and -1, however, I don't know how to arrive at the solution.

    Can it be done using simple log laws?
    The equation...

    f(x)=- 2^{x}-x=0 (1)

    ... cannot be solved using simple log laws!... once You have found in graphical way that the solution is between -1 and 0, if You need better accuracy the only choice You have are numerical methods. Among them one of the most simple [and elegant...] even if not widely used is to consider the solution as the limit [if it exists...] of the sequence defined by the recursive relation...

    \Delta_{n}= x_{n+1}-x_{n}= f(x_{n}) (1)

    In Your case the (1) is...

    x_{n+1}= -2^{x_{n}} (2)

    What You have to do now is to compute iteratively the x_{n} till to an n 'large enough'. Starting with x_{0}=0 You easily find...

    x_{1}= -2^{0}= -1

    x_{2}= -2^{-1}= -.5

    x_{3}= -2^{-.5}= -.707106...

    x_{4}= -2^{-.707106...}= - .612547...

    x_{5}= -2^{-.612547...}= -.65404...

    x_{6}= -2^{-.65404...}= -.6354978...

    x_{7}= -2^{-.6354978...}= -.6437186...

    x_{8}= -2^{-.6437186...}= -64006102...

    ... and so one. The convergence to the solution x \sim -.641186 is evident...

    Kind regards

    \chi \sigma
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  7. #7
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    Re: Simple equation help?

    Quote Originally Posted by eskimogenius View Post
    Hi there,

    This popped up in my exam the other day, absolutely no idea how to solve it using secondary school knowledge:

    2^x = -x
    Clearly you are not expected to solve it by hand, for the simple fact that it cannot be solved exactly by hand or otherwise (unless you use a special function called the Lambert W-function).

    Did you have a graphics or CAS calculator in the exam? You are expected to know how to use your calculator to solve it.
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