Logarithm problem

[goushujin]

How do you solve equations like:
ln (x) = ax
where a is any positive whole number

[TheManWhoCan]

You can't do it algebraically so you'd have to use the Newton-Rhapson method which goes something like this:

f(x) = ln(x) - ax
f'(x) = 1/x - a

x_r+1 = x_r - (ln(x_r) - ax_r)/(1/x_r - a)

Depending on the value of a, the equation f(x)=0 will have zero, one or two solutions. There's no foolproof way to find them other than sticking random numbers for x_r into that equation to find the better approximation x_r+1 until x_r+1 converges to a specific value.