14 thoughts on “A little math puzzle

  1. x/log(x)=100
    And the other solution is one very close to 1 from infinitity; that’s another number such that its logarithm is 1/100 of its value. Below 1 it would be negative, and zero and below it would not be defined. (I’m using non-calculus high school math here.)

  2. The basic equation is log(100x) = x. This simplifies to log(100) + log(x) = x, and thus 2 + log(x) – x = 0.

    Let y = 2 + log(x) – x. We are looking for the places where this function intercepts the x axis. x=2.3758… is one of them. This function is undefined for x0, y approaches negative infinity.

    Take the first derivative, and you get y’ = (1/x) – 1. When x1, it’s negative, and the slope is decreasing.

    So, y starts at negative infinity when x is close to 0, increases until x=1, then decreases after that. Since one x-intercept is at a value of x that is greater than the point at which the function is maximized, there must be some point 0<x<1 at which the function intersects the x-axis. This will be the only other solution. It's close to x=0.01024.

  3. x = log(100*x)
    -> x = log(100) + log(x)
    -> x = log(x) + 2

    This gives the second solution as being slightly above 0.01. There cannot be more than two solutions to this equation because the left-hand side is a straight line and the right-hand side is strictly concave everywhere.

  4. Not my most elegant proof, but it’s been a long time since I’ve done this.

    The problem is this: log(x) = x/100

    1) Both log(x) and x/100 are continuous over the set of all positive numbers.
    2) Log(x) is a curve that always has a negative (and decreasing) curvature.
    3) because curvature always has the same sign, a straight line can only intersect it once, twice, or not at all.
    4) Because x/100 is a straight line, it may not intersect log(x) more than twice
    5) for values of x=0 and x=infinity, x/100 is greater than log(x)
    6) For some values of x (example 10), x/100 is less than log(x)
    7) From this we can conclude that
    a) The x/100 and log(x) must intersect
    b) they must intersect twice.

    So, if 2.3…. is a solution, there is exactly one other solution.

  5. For Nemo’s bonus puzzle, the answer is

    1/ln(10)+log10(ln(10))

    which is about 0.79651017.

  6. The number 2.375812087593 is the root of the following equation:

    x^2- 3.375812087593x + 2.375812087593 = 0

    The only other solution appears to be 1. What do I win?

  7. Yes, that’s a nicer way to write it. I just wrote down what I got from solving it, which gave me that ugly mess of logs of different bases…

  8. Fascinating, did no one else immediately see the generalization to x = log x + n, where n is any integer? Obviously that doesn’t have just one other solution, but it makes a very neat pattern. There are two series of solutions, one series which quickly approaches 1, and one which grows a little faster than n. The easiest way to see the whole thing is to look at the graph of x – log(x).

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