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Chapter 3: Problem 106

Describe the quotient rule for logarithms and give an example.

Short Answer

Expert verified
In essence, the quotient rule for logarithms states that \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\). This implies that division inside the logarithm can be converted to subtraction outside of it. A clear instance of this is \(\log_2(\frac{8}{4})\), which simplifies, via the quotient rule, to \(\log_2(8) - \log_2(4)\) equals 1.

Step by step solution

01

Defining the Quotient Rule

The Quotient Rule for logarithms states that for any two positive real numbers a and b such that \(b \neq 1\), and any real number r, the logarithm base b of a quotient is the difference of the logarithms. It can be mathematically formulated as follows: \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\).
02

Explaining the Quotient Rule

This rule essentially says that division inside the logarithm can be turned into subtraction outside the logarithm, and vice versa. The numerator of the fraction inside the logarithm corresponds to the first part of the subtraction outside the logarithm, and the denominator of the fraction inside the logarithm corresponds to the second part of the subtraction outside the logarithm.
03

Example of the Quotient Rule

For illustration, an example is provided. If we have \(\log_2(\frac{8}{4})\), by applying the quotient rule for logarithms, this expression simplifies to \(\log_2(8) - \log_2(4)\), which further simplifies to 3 - 2, hence the result is 1.

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