Solving for n involving logarithm

Question

I know how to solve if the left side is just lgn, but it has an extra n. Was it correct to multiply both sides by e?

Answer 1

That is not the correct way to do it. Indeed, checking your answer into the original line, it is obvious this is false. Instead, note the Lambert W function:

$$n\ln(n)=\ln(n)e^{\ln(n)}=10^6\ln(2)$$

$$\implies\ln(n)=W_0(10^6\ln(2))$$

$$\large n=e^{W_0(10^6\ln(2))}$$