Find root of the equation Find maximum root of the equation $$x - \frac{1000}{\log 2} \log x = 0$$
It locates between $13746$ and $13747$, but I want to find right solution not using graphing calculators. Thanks in advance.
 A: Consider function $$f(x)=x - \frac{1000}{\log 2} \log (x)$$ $$f'(x)=1-\frac{1000}{x \log (2)}$$ The function goes through a minimum (by second derivative test) at $x=\frac{1000}{\log (2)}$.
So, let us start Newton method which will generate the following iterates $$x_0=2000$$ $$x_1=34175.5$$ $$x_2=14218.2$$ $$x_3=13747.7$$ $$x_4=13746.8$$ which is the solution for six significant figures.
Edit
What I tried to show above is that, even with a very poor estimate of the solution, we can get the solution in few iterations of Newton method. However, we can improve the process since, for large values of $k$, the root of equation $f(x)=x-k\log(x)=0$ can be estimated as $$x\approx k\Big(\log(k)+\log\big(\log(k)\big)\Big)$$ For $k= \frac{1000}{\log 2}$, this will give $x_0= 13357.4$ and the solution would have been obtained after two iterations.
A: Solve $$x-\frac{1000}{\log(2)}\log(x)=0$$ for $x$.
Substitute $x=e^t$:
$$e^t-\frac{1000}{\log(2)}t=0;$$
subtract $e^t$ from both sides:
$$-\frac{1000}{\log(2)}t=-e^t;$$
multiply both sides by $\dfrac{\log(2)}{1000}$:
$$-t=-\frac{\log(2)}{1000}e^t;$$
divide both sides by $e^t$:
$$-t\frac{1}{e^t}=-\frac{\log(2)}{1000};$$
rewrite $1/e^t=e^{-t}$:
$$-te^{-t}=-\frac{\log(2)}{1000};$$
substitute $u=-t$:
$$ue^u=-\frac{\log(2)}{1000};$$
take the branch $-1$ product log of both sides:
$$u=\operatorname{W}_{-1}\left(-\frac{\log(2)}{1000}\right);$$
substitute back for $u$:
$$-t=\operatorname{W}_{-1}\left(-\frac{\log(2)}{1000}\right);$$
multiply both sides by $-1$:
$$t=-\operatorname{W}_{-1}\left(-\frac{\log(2)}{1000}\right);$$
substitute back for $t$:
$$\log(x)=-\operatorname{W}_{-1}\left(-\frac{\log(2)}{1000}\right);$$
take exponentials of both sides:
$$x=e^{-\operatorname{W}_{-1}\left(-\log(2)/1000\right)}\phantom{.};$$
write the answer out:
$$\therefore x=\boxed{e^{-\operatorname{W}_{-1}\left(-\log(2)/1000\right)}\phantom{.}}\approx13746.809166647028808721383407435.$$
A: $$x-\frac{1000\ln(x)}{\ln(2)}=0\Longleftrightarrow$$
$$-\frac{1000\ln(x)}{\ln(2)}=-x\Longleftrightarrow$$
$$\frac{1000\ln(x)}{\ln(2)}=x\Longleftrightarrow$$
$$1000\ln(x)=x\ln(2)\Longleftrightarrow$$
$$e^{1000\ln(x)}=e^{\ln(2)x}\Longleftrightarrow$$
$$x^{1000}=2^x\Longleftrightarrow$$
$$x=\exp\left[-\text{W}\left(-\frac{\ln(2)}{1000}\right)\right]\approx13746.809166647028809$$
