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?

@SimplyBeautifulArt OK, that's what I assumed from the context. Is it more common outside the US? Or is it somewhat esoteric? — The Count, Jan 30 '17 at 01:48
There is no solution in elementary functions for a problem such as this. You might want to look at the Lambert W-function — rogerl, Jan 30 '17 at 01:50

score 0 · Answer 1 · answered Jan 30 '17 at 01:50

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:

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

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

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

Solving for n involving logarithm

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