136k views
2 votes
If log₄ 9 = a and log₂₇ 5 = b, express log₃ 80 in terms of a and b.

a) (5/2)a - (3/4)b))
b) (3/2)a + (4/5)b))
c) (3/2)a - (4/5)b))
d) (5/2)a + (3/4)b))

User Mahmudul
by
7.8k points

1 Answer

1 vote

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.

User Yasmuru
by
8.1k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories