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How do I solve this equation?

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

I know it has solution (saw graph).

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  • $\begingroup$ For an exact answer, you have to use the Lambert W function. Otherwise, you can approximate it using Newton's method. $\endgroup$
    – Toby Mak
    Commented Apr 22, 2020 at 10:55

1 Answer 1

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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!

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