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        riddles >> medium >> log (100*x) = x
        
(Message started by: Christine on Jun 16^th, 2013, 11:43am)

Title: log (100*x) = x
Post by Christine on Jun 16^th, 2013, 11:43am

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?

Title: Re: log (100*x) = x
Post by towr on Jun 16^th, 2013, 10:43pm

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)

Title: Re: log (100*x) = x
Post by SWF on Jul 5^th, 2013, 9:53pm

An answer may be expressed in terms of the Lambert W function:
x=10^x-k
x/10^x=1/10^k=x*exp(-x*ln(10))
(-x*ln(10))*exp(-x*ln(10))=-ln(10)/10^k
This equation has the form expressible by the Lambert W function (this provides both solutions since W() can be multivalued)
W(-ln(10)/10^k)=-x*ln(10)

x=-W(-ln(10)/10^k)/ln(10)