How do I convince students in high school for which this equation: $2^x=4x$ have only one solution in integers that is $x=4$?

Hint:

plot the graph of $y=2^x$ and $y=4x$ and shows that the only other solution is between $0$ and $1$.


If your intention is to 'convince' and not to prove, I'd draw a graph with the functions $y=2^x$ and $y=4x$. The growth rate of each function should make clear that they intersect only at two points, being the first between $0$ and $1$ (and hence, not being an integer).


For positive $n$, we have two growing sequences

$$1,2,4,8,\color{green}{16},32,64,128,256\cdots\\ 0,4,8,12,\color{green}{16},20,24,28,32\cdots$$

This shows them that the "curves" cross each other at $16$, and it seems that the first grows faster.

Indeed, taking the ratios of successive terms

$$\frac{2^{n+1}}{2^n}=2>\frac{4(n+1)}{4n}=1+\frac1n.$$

Tags:

Exponentiation

Education

Real Analysis

Logarithms

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