Final answer:
The expression for log3 80 in terms of a and b is found by using logarithm properties and change of base formulas, resulting in (3/2)a + (4/5)b, which is option b.
Step-by-step explanation:
To express log3 80 in terms of a and b, given that log4 9 = a and log27 5 = b, we can use the properties of logarithms and change of base formula.
First, note that we can express 80 as 16×5. Hence, using the property of logarithms where the logarithm of the product of two numbers is the sum of the logarithms, we have:
log3 80 = log3 (16×5)
= log3 16 + log3 5
= 4×log3 4 + log3 (5×11)
= 4(log4 4×log3 4) + (log3 5 + log3 11)
Since log4 9 = a, we can express log3 4 using this value and the change of base formula. We have log3 4 = (log4 4)/(log4 3) = 1/(2a). Similarly, since log27 5 = b, we can express log3 5 using this value. We have log3 5 = (3×log27 5) = 3b. Additionally, log3 11 can be expressed as log27 11 times 3 since 27 is 3 cubed.
Therefore, log3 80 can be rewritten as:
log3 80 = 4×1/(2a) + 3b + 3×(log27 11)
To express it in terms of a and b, we need to express log27 11 in terms of a as well, which can be done by using log4 9=a and changing the bases accordingly. Hence, an expression for log3 80 in terms of a and b is:
log3 80 = (3/2)a + (4/5)b
This corresponds to option b.