Solve for $x$: $2^x=4x$

Given that $x$ is a positive integer. By using methods of trial and error as well as plotting two lines: $y=2^x$, $y=4x$ on a graph and find their intersection point, we can easily solve for $x$ which is equal to 4. However, I do not know how to solve this using only equation. Can anyone help me?

Solution 1:

Prove that $2^x>4x$ for $x > 4$ by induction. It's very easy since: $2^{x+1}>4(x+1)$ equivalent to $2^x + 2^x> 4x + 4x$

Solution 2:

In the first place, $4$ is a solution but not the only solution. There is still another solution that lies at about $0.31$.

In the second place, there is no effective way to find analytic solutions for this equation using only the equation. As mentioned in the comments, $4$ is a good integer, we stumble on it and find it just fits exactly good. But $0.309\cdots$ is a "bad" number and there is no possibility we'd stumble on it.

However there is a numerical way that we can approach the $0.309\cdots$ solution as close as we desire. To do this, we can make use of the contraction mapping principle. Just note that $\phi(x)=2^x/4$ is a contraction on the Banach space $[0,2^{4/\ln2}]$ (you can check the Lipschitz constant), you can arbitrarily pick a $x_0$ within this interval and get the (only) fixed point $x*$ via iteration: $$x_n=\phi(x_{n-1}),\,x*=\lim_{n\to\infty}x_n$$ which is just the desired solution.