I don't understand how log2 √(1/2) turned into log2 2^(-1/2).

Quote:
You will have to know the 3 prime properties of logs

1. logk (AB) = logk A + logk B
2. logk(A/B) = logk A - logk B
3. logk (A^n) = n logk A
where k is any positive number , k ≠ 1

so log2√36 - log2 log2</sub√72
= log2 (√36/√72)
= log2 √(36/72)
= log2 √(1/2)
= log2 2^(-1/2)
= (-1/2) log2 2
= (-1/2)(1)
= -1/2

Review the laws of exponents

First of all √2 = 2^(1/2)
then
√(1/2)
= √1/√2
= 1/√2 = 1/2^(1/2) = 2^(-1/2)

based on 1/a^n = a^-n

To understand how log₂ √(1/2) turned into log₂ 2^(-1/2), let's break it down step by step:

1. Start with log₂ √(1/2).
2. Recall that the square root of a number can be written as that number raised to the power of 1/2. So, we can rewrite √(1/2) as (1/2)^(1/2).
3. Using the property of logarithms, logₐ (x^y) = y * logₐ (x), we can bring the exponent 1/2 out in front of the logarithm. This property allows us to break down the expression and simplify it further.
4. Applying the property, we have log₂ ((1/2)^(1/2)) = (1/2) * log₂ (1/2).
5. Now, we need to simplify log₂ (1/2). To do this, we can use another property of logarithms, logₐ (1/x) = -logₐ (x). This property states that the logarithm of the reciprocal of a number is equal to the negative logarithm of that number.
6. Applying the property, log₂ (1/2) = -log₂ 2.
7. Finally, we arrived at -log₂ 2. Notice that log₂ 2 is equal to 1 because 2 raised to what power gives us 2? The answer is 1. So, we can simplify -log₂ 2 as -1.
8. Combining all the simplifications, we have (-1/2) * 1 = -1/2.

Therefore, log₂ √(1/2) simplifies to -1/2 using the properties of logarithms and exponents.