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

Question

Let
log
b
m
=
x
and
log
b
n
=
y
.

log
b
m
=
x
is equivalent to
b
x
=
m
.

log
b
n
=
y
is equivalent to
b
y
=
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
m
n
=
b
x
b
y
.
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 product of m and n as
m
n
=
b
x
b
y
.
Write the product of m and n as m n equals b superscript x baseline b superscript y baseline .

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

Write the sum of m and n as
m
+
n
=
b
x
+
b
y
.
Write the sum of m and n as m plus n equals b superscript x baseline plus b superscript y baseline .
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Answer 1

To prove the Quotient Rule of Logarithms, you want to relate the logarithms of the numbers \( m \) and \( n \) by considering their quotient \( \frac{m}{n} \).

You have:

• \( \log_b m = x \) implies \( b^x = m \).
• \( \log_b n = y \) implies \( b^y = n \).

The next logical step is to express the quotient \( \frac{m}{n} \) in terms of \( b^x \) and \( b^y \). Therefore, you should write:

\[ \frac{m}{n} = \frac{b^x}{b^y} \]

This corresponds to the following option:

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