Expert: David Hemmer - 7/13/2007

Question
how would one solve deductively an equation in one variable where x occurs both in a logarithm and an exponential, such as in the equation x=8*log_2(x)

Answer
There is no good way to do this, i.e. the solution does not have an answer in closed form. To write the solution you need to use something called the LambertW function, see

http://en.wikipedia.org/wiki/Lambert's_W_function

for information.

Using Maple I solved your equation and got the two solutions:

-8/ln(2)*LambertW(-1/8*ln(2)), -8/ln(2)*LambertW(-1,-1/8*ln(2))

Of course if you just want an approximate solution, you could just plot x-8*log_2(x) on a graphing calculator and get an estimate for where the graph intersects the x axis.


Consider for instance the simpler equation

x=log x.

Even this equation does not have a "closed form" solution for x.