How to solve this equation? $t-ln|t+1|+1 = 0$

Question

How do I solve this equation?

$$t-ln|t+1|+1 = 0$$

I know it has solution (saw graph).

Answer 1

To solve this equation, write it as ln|t+1|= t+1 and let x= t+ 1 so that it becomes ln|x|= x. Now take the exponential of both sides to get $|x|= e^x$ or $|x|e^{-x}= 1$ Finally let y= -x so the equation can be written $-ye^y= 1$ or $ye^y= -1$.

Now to try to solve that you could try. as Toby Mak suggested, either a numerical method or "Lambert's W function" which is defined as the inverse function to $f(x)= xe^x$.

OUCH! I forgot the absolute value! The equation is $ye^y= 1$, not -1 so has a solution!