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Topic: log (100*x) = x (Read 2606 times) |
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Christine
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log (100*x) = x
« on: Jun 16th, 2013, 11:43am » |
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Futility closet gives:
http://www.futilitycloset.com/2011/02/28/misc-22/
log 237.5812087593 = 2.375812087593
which is false.
Can you find x so that log(100*x) = x
that is
log(100) + log(x) = x
2 + log(x) – x = 0
And in general
For what k does x = log 10(x) + k has a solution?
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towr
wu::riddles Moderator
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Re: log (100*x) = x
« Reply #1 on: Jun 16th, 2013, 10:43pm » |
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There are solutions for all k >= 1, the easiest way to see that is to draw the graph of log(x) +k and x and see when they cross.
Also, log10(237.5812087593) = 2.375812087593 is correct (as approximation, since both number go on forever) And there is another solution (as you could see from the graph)
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Wikipedia, Google, Mathworld, Integer sequence DB
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SWF
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Re: log (100*x) = x
« Reply #2 on: Jul 5th, 2013, 9:53pm » |
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An answer may be expressed in terms of the Lambert W function:
x=10x-k
x/10x=1/10k=x*exp(-x*ln(10))
(-x*ln(10))*exp(-x*ln(10))=-ln(10)/10k
This equation has the form expressible by the Lambert W function (this provides both solutions since W() can be multivalued)
W(-ln(10)/10k)=-x*ln(10)
x=-W(-ln(10)/10k)/ln(10)
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