# Solving a logarithmic equation with variables on each side. Okay, so while doing a problem for my calculus class I was required to graph two functions in order to see where they intersect, as according to my teacher there is no way to solve it analytically. This really bothers me and there must be a way to solve it. I did research online to try to solve it, but I had no idea where to even start. Here is the equation. y = ln(5 - y)

Solving a logarithmic equation with variables on each side
Okay, so while doing a problem for my calculus class I was required to graph two functions in order to see where they intersect, as according to my teacher there is no way to solve it analytically.
This really bothers me and there must be a way to solve it. I did research online to try to solve it, but I had no idea where to even start.
Here is the equation.
$y=\mathrm{ln}\left(5-y\right)$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

fjaldangi
For this one you need Lambert's Omega function which is defined as:
$W\left(z\right){e}^{W\left(z\right)}=z$
Your equation can be written as:
$x=\mathrm{ln}\left(5-x\right)$
$x=\mathrm{ln}\left(5-x\right)$
${e}^{x}+x=5$
Which solution is $x=5-W\left({e}^{5}\right)$