Final answer:
The product of logb(2/3) and logb(3/2) is zero, which is derived using the properties of logarithms to simplify the expression.
Step-by-step explanation:
The question is asking to find the value of logb(2/3) multiplied by logb(3/2). Using the properties of logarithms, we can find this product. According to the property that states the logarithm of a division is the difference between the logarithms of the numerator and the denominator, we determine that:
logb(2/3) = logb(2) - logb(3)
In the same manner, we can express:
logb(3/2) = logb(3) - logb(2)
When we multiply these two logarithms together, we get:
logb(2/3) × logb(3/2) = (logb(2) - logb(3)) × (logb(3) - logb(2))
Expanding this expression, we get a difference of squares:
(logb(2))2 - (logb(3))2
However, since we are subtracting the square of the same quantities but in reverse order, this simplifies to:
-((logb(3))2 - (logb(2))2)
As we apply the difference of squares formula (a2 - b2 = (a+b)(a-b)), we realize the terms within the parenthesis are the negation of each other, leading to:
-(0) = 0
Therefore, the product of the two logarithms is zero. This demonstrates the utility of the properties of logarithms in simplifying expressions and solving problems.