Logarithm problem that I'm stuck on

Gold Member
The problem statement
ln(x) = 5 -x
Solve for x.

The attempt at a solution
$$ln x = 5 - x \\ e^{ln x} = e^{5 - x} \\ e^{ln x} = \frac{e^5}{e^x} \\ x e^x = e^5$$
Here is the place where I get stuck.

DrDu
There is no analytical solution. You have to solve it numerically, e.g. ##x_{n+1}=5-\ln(x_n)## starting with ##x_0=5##.

Orodruin
Staff Emeritus
Homework Helper
Gold Member
In addition to what DrDu said already, the formal solution to the problem is given in terms of the Lambert W function. As indicated in the previous post, you will have to find the solutions to this numerically.

##5 = x + ln(x)##

Consider:
##y = 5##
and
##y = x + ln(x)##

Take an educated guess at a solution.

##y_n = x_n + ln(x_n)##
_
Consider the slope of the tangent line to this function at ##x_n##.

The slope of the tangent line of this function at ##x_n## is approximately equal to the slope of the secant line of this function between ##x_n## and ##x_{n+1}##.

Let ##y_{n+1} = 5##.

Find your new ## x_{n+1}##.

Repeat iteratively.

Within 3 iterations (by hand), I converged on the solution.

HallsofIvy
The "Lambert W function" that Orodruin referred to is defined as the inverse function to $f(x)= xe^x$. So, immediately, x= W(e^5).