Problem 107
Question
Describe the power rule for logarithms and give an example.
Step-by-Step Solution
Verified Answer
The power rule for logarithms states that \( \log_b (b^n) = n \cdot \log_b (b) \). For example, \( \log_2 (2^3) \) simplifies to 3.
1Step 1: Defining the Power Rule for Logarithms
The power rule for logarithms states that for any real numbers \( b \) and \( x \), and integer \( n \), the logarithm of \( b^n \) can be written as the product of \( n \) and the logarithm of \( b \). Formally, this is: \( \log_b (b^n) = n \cdot \log_b (b) \)
2Step 2: Example of Applying the Power Rule for Logarithms
As a practical example, consider \( \log_2 (2^3) \). Using the power rule for logarithms, this expression simplifies to \( 3 \cdot \log_2 (2) \). Since the log base 2 of 2 is 1, the final answer is \( 3 \cdot 1 = 3 \).
Other exercises in this chapter
Problem 106
Describe the quotient rule for logarithms and give an example.
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
In Exercises 105–108, evaluate each expression without using a calculator. $$ \log _{2}\left(\log _{3} 81\right) $$
View solution Problem 108
Without showing the details, explain how to condense \(\ln x-2 \ln (x+1)\)
View solution