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Chapter 4: Problem 106

Describe the quotient rule for logarithms and give an example.

Short Answer

Expert verified
The quotient rule for logarithms states that the log of a division equals the log of the numerator subtracted by the log of the denominator. For instance, \( log_2(\frac{8}{2}) = log_2(8) - log_2(2) \), which simplifies to \( 2 \).

Step by step solution

01

Definition of Quotient Rule for Logarithms

The Quotient Rule for logarithms states that the logarithm of a quotient is the difference of the logarithms. This can be mathematically expressed as: \( log_b(\frac{m}{n}) = log_b(m) - log_b(n) \) where \(b\) is the base of the logarithm, and \(m\) and \(n\) are the quantities being divided.
02

Explanation of the Rule

In simple terms, this rule expresses that the log of a division equals the log of the numerator subtracted by the log of the denominator.
03

Providing an Example

Let's provide an example to understand this rule better. Consider the logarithmic expression: \( log_2(\frac{8}{2}) \). Using the quotient rule, it can be broken down as follows: \( log_2(8) - log_2(2) \). By simplifying, we have \( 3 - 1 = 2 \). Hence, \( log_2(\frac{8}{2}) = 2 \), proving the quotient rule for logarithms.

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