 How do I convince students in high school for which Helen Lewis 2022-01-04 Answered
How do I convince students in high school for which this equation: $$\displaystyle{2}^{{x}}={4}{x}$$ have only one solution in integers that is $$\displaystyle{x}={4}$$?

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Hint:
plot the graph of $$\displaystyle{y}={2}^{{x}}$$ and $$\displaystyle{y}={4}{x}$$ and shows that the only other solution is between 0 and 1.
Not exactly what you’re looking for? Jeffery Autrey
If your intention is to 'convince' and not to prove, I'd draw a graph with the functions $$\displaystyle{y}={2}^{{x}}\ \text{ and }\ {y}={4}{x}$$.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). Vasquez

For positive n, we have two growing sequences
1,2,4,8,16,32,64,128,256⋯
0,4,8,12,16,20,24,28,32⋯
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
$$\frac{2^{n+1}}{2^n}=2>\frac{4(n+1)}{4n}=1+\frac{1}{n}$$