How would you solve this problem using log?

Question

I have this equation and I want to find the possible values of $n$. So how would you solve this using logarithms?

$10n^2 = 2^n$

Answer 1

There is no way to find the solution of this equation using logarithm.

You can find the roots with some numerical method or using the Lambert W function. If you want a simple estimate of the roots tha you can sketch the graphs of the functions

$y =10n^2$ and $y=2^n$ ( it is easy)

and search the common points. You see immediately that there is a negative solution $x_1<0$ and a positive solution $0<x_2<1$ because $10\cdot 1^2 > 2^1$. Since we have $2^{10} < 10\cdot 10^2$ there is also another solution $1<x_3<10$ (and you can easily restrict the interval to $9<x_3<10$).