log2√36 - log2√72

Question

log2√36 - log2√72

I am not sure how to do this question. I need an explanation.

Answer 1

You will have to know the 3 prime properties of logs

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

so log₂√36 - log₂ log_{2</sub√72

= log₂ (√36/√72)

= log₂ √(36/72)

= log₂ √(1/2)

= log₂ 2^(-1/2)

= (-1/2) log₂ 2

= (-1/2)(1)

= -1/2}

Answer 2

log(√x) = 1/2 log(x)

since log(x^n) = n log(x)

So, what you have is
√36 = 6, so

√72 = 6√2

So, assuming by log2 you mean log₂,

log₂6 + log₂6 + log₂2
but log₂2 = 1, so that is
= 2log₂6 + 1/2

Now, what is log₂6?
You don't have a log₂ button on your calculator, but you can easily get it online.

Or, you can change base, remembering that

log₂6 = log₁₀6/log₁₀2

Answer 3

Oops. I added, instead of subtracting. The final answer, of course, is -1/2, as shown above.

Answer 4

To solve this problem, you'll need to use the properties of logarithms. Specifically, you'll need to understand the properties of logarithms related to exponentiation and division.

Let's break down the question step by step:

1. Start with the expression: log2√36 - log2√72

2. Use the property of logarithms that states: logb(A) - logb(B) = logb(A/B)
Applying this property, we can rewrite the expression as: log2(√36/√72)

3. Simplify the expression inside the logarithm: √36 is equal to 6, and √72 can be simplified as √(36 * 2), which is equal to 6√2.
So, the expression simplifies to: log2(6/6√2)

4. Simplify further by canceling out the common factor: 6/6 is equal to 1, so the expression becomes: log2(1/√2)

5. Now, simplify the expression inside the logarithm: 1/√2 can be rationalized by multiplying the numerator and denominator by √2.
This gives us: log2(√2/2)

6. Using the property of logarithms that states: logb(A) - logb(B) = logb(A/B), we can rewrite the expression as: log2(√2) - log2(2)

7. Recall that the square root of 2 can be expressed as 2^(1/2) and 2 can be expressed as 2^1.
Therefore, the expression becomes: log2(2^(1/2)) - log2(2^1)

8. Apply the property of logarithms that states: logb(b^m) = m, to simplify the expression.
This gives us: 1/2 - 1

9. Finally, subtract: 1/2 - 1 = -1/2

Therefore, log2√36 - log2√72 is equal to -1/2.