You are proving the Quotient Rule of Logarithms. The following shows your work so far.

Question

Let logbm=x
and logbn=y
.

logbm=x
is equivalent to bx=m
.

logbn=y
is equivalent to by=n
.

Which of the following options best describes the next step you should take in this proof?

(1 point)
Responses

Write the quotient of m and n as mn=bxby
.
Write the quotient of m and n as Start Fraction m over n End Fraction equals Start Fraction b superscript x baseline over b superscript y baseline End Fraction .

Write the difference between m and n as m−n=bx−by
.
Write the difference between m and n as m minus n equals b superscript x baseline minus b superscript y baseline .

Write the product of m and n as mn=bxby
.
Write the product of m and n as m n equals b superscript x baseline b superscript y baseline .

Write the sum of m and n as m+n=bx+by
.

Answer 1

To prove the Quotient Rule of Logarithms, you want to manipulate the expressions for \( m \) and \( n \) based on their definitions in terms of logarithms. Since you have:

1. \( \log_b m = x \) which implies \( b^x = m \)
2. \( \log_b n = y \) which implies \( b^y = n \)

Next, since you are looking to show the relationship between the logarithms of the quotient \( \frac{m}{n} \), you would take the ratio of \( m \) and \( n \).

So, the best next step to take would be:

Write the quotient of m and n as \( \frac{m}{n} = \frac{b^x}{b^y} \).

This step will allow you to derive the logarithmic property that involves the difference, by simplifying \( \frac{m}{n} \) using the properties of exponents.