Question

Iog√2(log2 3•log3 4)

Answer 1

Решения на листочке:)

Answer 2

It seems like you're asking to simplify the expression:

$\log(\sqrt{2} \cdot (\log_2{3} \cdot \log_3{4}))$

To simplify this expression, you can use logarithm properties. Let's break it down step by step:

1. Start with the innermost expressions: $\log_2{3} \cdot \log_3{4}$

2. Use the change of base formula to express both logarithms in the same base (let's use the common base 10): $\log_2{3} = \frac{\log{3}}{\log{2}}$ $\log_3{4} = \frac{\log{4}}{\log{3}}$

3. Substitute these expressions back into the original expression: $\log(\sqrt{2} \cdot \left(\frac{\log{3}}{\log{2}} \cdot \frac{\log{4}}{\log{3}}\right))$

4. Simplify the expression inside the parentheses: $\log(\sqrt{2} \cdot \frac{\log{3} \cdot \log{4}}{\log{2} \cdot \log{3}})$

5. Use the properties of logarithms to further simplify: $\log(\sqrt{2} \cdot \frac{\log{3} \cdot \log{4}}{\log{2} \cdot \log{3}})$ $\log(\sqrt{2} \cdot \frac{\log{4}}{\log{2}})$

6. Now, you can use the properties of logarithms to simplify the expression further. Remember that $\log{a} + \log{b} = \log{(a \cdot b)}$: $\log(\sqrt{2}) + \log(\frac{\log{4}}{\log{2}})$

7. Since $\log(\sqrt{2})$ is the same as $\frac{1}{2} \log{2}$, we can rewrite it as: $\frac{1}{2} \log{2} + \log(\frac{\log{4}}{\log{2}})$

8. Use the properties of logarithms again to simplify $\log(\frac{\log{4}}{\log{2}})$: $\frac{1}{2} \log{2} + \log(\log{4}) - \log(\log{2})$

So, the simplified expression is: $\frac{1}{2} \log{2} + \log(\log{4}) - \log(\log{2})$