The value of log√2 (√2 × √2 × √2 × √2) is

Power rule of logarithms: logb(bn) = n, because raising the base to that same power is, by definition, exactly the exponent n.

Also, when several equal factors are multiplied together they combine into a single power: repeating a term k times equals that term raised to the power k.

1. Combine the four equal factors: √2 × √2 × √2 × √2 = (√2)4, since the same base multiplied by itself four times equals that base raised to the power 4.

2. The logarithm becomes log√2((√2)4).

3. Using the power rule logb(bn) = n, this equals 4 × log√2(√2).

4. Since logb(b) = 1 for any valid base greater than 0 and not equal to 1, log√2(√2) = 1.

5. Substituting: 4 × 1 = 4.

Independent check by switching to base 2: √2 equals 2 raised to the power one-half, so (√2)4 equals 22, i.e. 4. Then log√2(4) equals log2(4) divided by log2(√2), which is 2 divided by one-half, giving 4 again — the same result.

So the value of the expression is 4.