Question

Доказать, что (3)^log2(5) = (5)^log2(3)

Answer 1

возьмем корень степени log2(3):

3^(log2(5)/log2(3))=5

log2(5)/log2(3)=log3(5)

3^log3(5)=5

Равенство доказано

Answer 2

To prove that (3)^log2(5) = (5)^log2(3), we'll use the property of logarithms that states: log_a(b^c) = c * log_a(b).

Let's start by taking the logarithm of both sides of the equation:

log2[(3)^log2(5)] = log2[(5)^log2(3)]

Now, use the logarithm property mentioned earlier:

log2(5) * log2(3) = log2(3) * log2(5)

Now, we have log2(5) * log2(3) on both sides of the equation. To proceed with the proof, we can divide both sides by log2(5):

log2(5) * log2(3) / log2(5) = log2(3) * log2(5) / log2(5)

On the left side, log2(5) / log2(5) equals 1:

1 * log2(3) = log2(3) * 1

This simplifies to:

log2(3) = log2(3)

Since both sides of the equation are equal, the original statement is true:

(3)^log2(5) = (5)^log2(3)

So, we have successfully proved that (3)^log2(5) is indeed equal to (5)^log2(3).