Logarithm problem that I'm stuck on

  • #1
Rectifier
Gold Member
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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

  • #2
DrDu
Science Advisor
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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##.
 
  • #3
Orodruin
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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.
 
  • #4
212
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##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.
 
  • #5
HallsofIvy
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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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