Given that logbase2 a+logbase2 b=3 , calculate all the possible integer values of a if b is

also an integer value. Explain your reasoning.

log2 a + log2 b = 3

log2 (ab) = 3
ab = 2^3 = 8

using positive integer values, all possibilties for
(a,b) are
(1,8)
(2,4)
(4,2) and
(8,1)

testing one of them: a=4, b=2

LS = log24 +log2 2
= 2 + 1
= 3
= RS

1=24

yes

To find all the possible integer values of a given the equation log base 2 a + log base 2 b = 3, we can use the properties of logarithms.

Using the property of logarithms, log base b (a) + log base b (c) = log base b (a * c), we can rewrite the equation as log base 2 (a * b) = 3.

Since log base b (c) = n can be rewritten as b^n = c, we can rewrite the equation as 2^3 = a * b.

Simplifying further, we get 8 = a * b.

Now, we need to find all the possible pairs (a, b) such that their product is 8 and b is an integer.

The factors of 8 are (1, 8), (2, 4), and (4, 2), as well as (8, 1). So, the possible integer values of a are 1, 2, 4, and 8.

Therefore, the possible integer values of a are 1, 2, 4, and 8, when b is also an integer.

To solve this equation and find the possible integer values of a, we'll use the properties of logarithms. The equation given is:

logbase2 a + logbase2 b = 3

We can rewrite this using the logarithmic property that states: logbaseA (X) + logbaseA (Y) = logbaseA (X * Y). Applying this property, we can rewrite the equation as:

logbase2 (a * b) = 3

To solve for a, we need to isolate it. Since the base of the logarithm is 2, we can rewrite the equation in exponential form, which is the inverse of the logarithmic form. In exponential form, we have:

2^3 = a * b

Simplifying, we get:

8 = a * b

Now we need to find all possible integer values of a when b is also an integer. We can begin by listing all the factors of 8:

1 * 8 = 8
2 * 4 = 8
4 * 2 = 8
8 * 1 = 8

From these factor pairs, we can see that a can take on any of the possible integer values of 1, 2, 4, or 8 when b is also an integer.

Therefore, the possible integer values of a, given logbase2 a + logbase2 b = 3 and b is also an integer, are 1, 2, 4, and 8.