Question

Choose the letter of the expression listed on the right that completes each step to show how to use the power and product properties of logarithms to prove that the quotient property is true for logb x y . logb x y = = = = a logbx + logby-1 b logbx - logby c logbxy-1 d logbx - 1logby

Answer 1

To use the power and product properties of logarithms to demonstrate that the quotient property is true, we express y to the power of -1 and then split using the product property. This leads to the power property being used again to bring down the exponent, showing that logb(x/y) is indeed equal to logbx - logby.

Step-by-step explanation:

To demonstrate the quotient property of logarithms using the power and product properties, we start with the logarithm of a quotient, logb(x/y), and show it can be expressed as the difference of two logarithms. According to the quotient property, logb(x/y) = logbx - logby.

First, we use the power property of logarithms which states that the logarithm of a number raised to an exponent is the product of that exponent and the logarithm of the number; symbolically, logb(ac) = c ∙ logba. So, we can rewrite y as y1 and apply the power property to get:

logb(x/y) = logb(x ∙ y-1). From here, we apply the product property, that the logarithm of a product is the sum of the logarithms, to split the single logarithm into two:

logb(x ∙ y-1) = logbx + logb(y-1). Finally, we use the power property again to bring down the exponent -1:

logbx + logb(y-1) = logbx - logby.

Answer 2

C
A
D
B

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