Problem 106
Question
Describe the quotient rule for logarithms and give an example.
Step-by-Step Solution
Verified Answer
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 \( 2 \).
1Step 1: 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.
2Step 2: 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.
3Step 3: 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.
Other exercises in this chapter
Problem 105
Describe the product rule for logarithms and give an example.
View solution Problem 105
In Exercises 105–108, evaluate each expression without using a calculator. $$ \log _{3}\left(\log _{7} 7\right) $$
View solution Problem 106
In Exercises 105–108, evaluate each expression without using a calculator. $$ \log _{5}\left(\log _{2} 32\right) $$
View solution Problem 107
Describe the power rule for logarithms and give an example.
View solution