7. Solve the following problem using as many methods as possible. Methods are considered different if they use different mathematical ideas, as well as various technical approaches to implementing the same idea. Indicate the place of each method used in the school mathematics curriculum.

Compare the numbers $2+\log _{2} 6$ and $2 \sqrt{5}$.

## Sixth Olympiad (2012). I (correspondence) round
Solution 1. First option. We have,

$$
2+\log _{2} 6=3+\log _{2} 3>3+\log _{2} 2 \sqrt{2}=\frac{9}{2}>2 \sqrt{5}
$$

Alternatively, the estimates can be written as,

$$
2 \sqrt{5}-2<2 \cdot \frac{9}{4}-2=\frac{5}{2}<\log _{2} 4 \sqrt{2}<\log _{2} 6
$$

Topic: "Powers with Real Exponents. Logarithms"