Ask a New Question
Menu

You are proving the Quotient Rule of Logarithms. The following shows your work so far. Let logbm=x log 𝑏 ⁡ 𝑚 = 𝑥 and logbn=y log 𝑏 ⁡ 𝑛 = 𝑦 . logbm=x log 𝑏 ⁡ 𝑚 = 𝑥 is equivalent to bx=m 𝑏 𝑥 = 𝑚 . logbn=y log 𝑏 ⁡ 𝑛 = 𝑦 is equivalent to by=n 𝑏 𝑦 = 𝑛 . Which of the following options best describes the next step you should take in this proof? (1 point) Responses Write the product of m and n as mn=bxby 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=bx−by 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=bx+by 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 .

1 answer

To prove the Quotient Rule of Logarithms, we start from the equations \( b^x = m \) and \( b^y = n \), where \( \log_b m = x \) and \( \log_b n = y \).

The next logical step in this proof is to consider the quotient \( \frac{m}{n} \). To express \( \frac{m}{n} \) in terms of the exponents \( x \) and \( y \):

  1. We have \( m = b^x \) and \( n = b^y \).
  2. Therefore, we can write the quotient \( \frac{m}{n} \) as: \[ \frac{m}{n} = \frac{b^x}{b^y} \]

Using the properties of exponents, this simplifies to: \[ \frac{m}{n} = b^{x - y} \]

Thus, taking the logarithm base \( b \) of both sides leads us to: \[ \log_b \left(\frac{m}{n}\right) = x - y \]

This shows that: \[ \log_b m - \log_b n = \log_b \left(\frac{m}{n}\right) \]

This indicates that we could describe the next step as expressing the quotient \( \frac{m}{n} \).

Given the options, the best description of the next step you should take in this proof is:

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

However, since this is not one of the options, the closest option that closely relates to considering \( m \) and \( n \) together and is directly relevant to the rules of logarithms would be:

Write the difference between m and n as \( m - n = b^x - b^y \).

This step would not directly prove the quotient rule, but it acknowledges the adjustment from individual logarithmic identities to a unified expression necessary for this proof. If clarity were provided within the options regarding the context of working towards the quotient, it would guide us toward a direct conclusion.