Explanation:
To solve this problem, we need to use the property of logarithms which states that: log a to the base b the whole power n is equal to n times log a to the base b.

Given:

log 2 to the base √x the whole square = log 2 to the base x

Solution:

Let's simplify the expression on the left-hand side of the equation using the property of logarithms mentioned above:

log 2 to the base √x the whole square = 2(log 2 to the base √x)

Now, we can rewrite the original equation as:

2(log 2 to the base √x) = log 2 to the base x

Using the property of logarithms again, we can simplify the left-hand side of the equation:

2(log 2 to the base √x) = log 2 to the base (√x) squared

2(log 2 to the base √x) = log 2 to the base x

log 2 to the base (√x) squared = log 2 to the base x

Now we can equate the expressions inside the logarithms:

(√x) squared = x

x = (√x) squared

Taking the square root of both sides, we get:

√x = x^(1/2)

x^(1/2) = x^(2/2)

x^(1/2) = x

Squaring both sides, we get:

x = x^2

Solving for x, we get:

x = 0 or x = 1

However, x cannot be equal to 0 because the square root of 0 is undefined. Therefore, the only solution is:

Answer: x = 1