Thursday, 2 July 2015

algebra precalculus - Find x when the function equals 0




I must solve for x for this function.



$e^x-20x=0$



I'm not sure what to do here. I've tried this so far but it makes no sense:
$$\begin{align*}
e^x&=20x\\
x\ln e&=\ln20+\ln x\\
\frac{x}{\ln20}&=\ln x

\end{align*}$$



I tried this as well but I'm not sure if this is right either:
$$\begin{align*}
e^x&=20x\\
\ln e^x&=\ln20x\\
x&=\ln20+\ln x
\end{align*}$$



I've got more confidence in the second try, although I don't know how to solve for an actual number.




The answer is a decimal.


Answer



This equation cannot be solved in terms of elementary functions. You can either use numeric methods like Newton's, or solve in terms of Lambert W function:



\begin{align*}
e^x - 20x &= 0 \\
20x &= e^x \\
xe^{-x} &= \frac{1}{20} \\
-x e^{-x} &= -\frac{1}{20} \\

-x &= W\left(-\frac{1}{20}\right) \\
x &= -W\left(-\frac{1}{20}\right)
\end{align*}


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