How would I answer this question?

Question

If log₂a + log₂b = 3 calculate all the possible integer values of a if b is also an integer value. Explain reasoning.

Answer 1

log₂a + log₂b = 3

log₂ab = 3

2^3 = 8 = ab

a = 1, 2, 4, or 8

b = 8, 4, 2 or 1

Answer 2

log₂ + log₂b = 3

log₂ ab = 3 Law of Logarithms

2² = ab Relation of logs to exponent

8 = ab

What are the positive integer factors of 8?

Answer 3

The log of a product is the sum of the logs, so

log₂(ab) = 3

Next exponentiate both sides with 2 to get

ab = 8 = 2³

We are limited to positive integers for both a and b in order to avoid taking log of a negative number or zero.. a and b must be powers of 2 to get 2³ on the right hand side

So a = 1, 2, 4 or 8