sove for x: lnx+4x=0?

Answer 1

There is no algebraic solution, or a solution in terms of common "simple" functions. As the previous answerer indicated, you can easily find a solution numerically, though.

To a few more decimal places, I get:

x = 0.3005419683031807

You can, however, express the solution in terms of a special function known as "Lambert's W function" (see source).

Lambert's W function is defined as the inverse function of
f(W) - W*exp(W). That is, if you have an equation:

Z = X*exp(X),

then X = W(Z),

where W(Z) is Lambert's function. The source below gives an algorithm for calculating numerical values of W(Z).

To use W(Z) in this case, we need to make your equation look like an equation of the form Z = X*exp(X). Starting with:

ln(x) + 4x = 0
ln(x) = -4x

exponentiate both sides:

x =exp(-4x)
x*exp(4x) = 1
4x*exp(4x) = 4

This is of the desired form, so

4x = W(4)
x = (W(4))/4

Answer 2

0.3005419683

Determined iteratively using Microsoft Excel. There are more digits in the actual solution (after the '3'). Maybe another answerer will come up with more digits.

.