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Logarithm problem that I'm stuck on

Rectifier
Gold Member
313
4
The problem statement
ln(x) = 5 -x
Solve for x.

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

Answers and Replies

DrDu
Science Advisor
6,012
749
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
Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
16,135
6,068
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.
 
212
49
##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
Science Advisor
Homework Helper
41,732
893
The "Lambert W function" that Orodruin referred to is defined as the inverse function to [itex]f(x)= xe^x[/itex]. So, immediately, x= W(e^5).
 

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