Problem 21
Question
In this Study Set, assume that all variables represent positive numbers and \(b \neq 1\). Evaluate each expression. See Example \(1 .\) $$ \log _{4} 4^{7} $$
Step-by-Step Solution
Verified Answer
The value of \( \log_{4} 4^{7} \) is 7.
1Step 1: Understand the Problem
You're given the mathematical expression \( \log_{4} 4^{7} \). It involves a logarithm with base 4 and an exponentiation of base 4.
2Step 2: Apply the Logarithm Exponent Rule
According to the logarithm property \( \log_{b} b^{x} = x \), where \( b \) is the base of both the logarithm and the exponent. Here, \( b = 4\) and \( x = 7 \), so \( \log_{4} 4^{7} = 7 \).
3Step 3: Verify the Solution
The expression simplifies exactly as expected: \( \log_{4} 4^{7} = 7 \). This is because logarithms essentially "undo" exponentiation when the base matches the base of the exponent.
Key Concepts
ExponentiationLogarithm PropertiesBase of Logarithm
Exponentiation
Exponentiation is a fundamental mathematical operation where a number, called the base, is raised to the power of an exponent. When you see a term like \( 4^7 \), it's telling us we need to multiply 4 by itself 7 times.
- **Base**: This is the number that is being multiplied. For \( 4^7 \), the base is 4.
- **Exponent**: This tells us how many times the base is used in a multiplication. In \( 4^7 \), the exponent is 7, so we multiply 4 by itself 7 times.
Exponentiation can grow values very quickly and is the opposite operation of taking a logarithm, which we will dive into next.
- **Base**: This is the number that is being multiplied. For \( 4^7 \), the base is 4.
- **Exponent**: This tells us how many times the base is used in a multiplication. In \( 4^7 \), the exponent is 7, so we multiply 4 by itself 7 times.
Exponentiation can grow values very quickly and is the opposite operation of taking a logarithm, which we will dive into next.
Logarithm Properties
Logarithms have unique properties that make them a handy mathematical tool, especially for reversing exponentiation. One crucial property is that they help us find the power to which a base number had to be raised to obtain another number.
- **Logarithm and Exponent Relationship**: If you have a logarithm \( \log_b a \), this essentially asks: "To what power must the base \( b \) be raised to result in \( a \)?"
- For example, \( \log_4 64 \) is 3 because \( 4^3 = 64 \).
- **Logarithm Exponent Rule**: One important rule is \( \log_b b^x = x \), meaning if the base of the logarithm and the base of the exponentiation are the same, the logarithms "cancel out" exponentiation, leaving you with the exponent.
- Therefore, for \( \log_4 4^7 \), based on this property, the result is simply 7, illustrating the cancelling effect because the base in both cases is 4.
- **Logarithm and Exponent Relationship**: If you have a logarithm \( \log_b a \), this essentially asks: "To what power must the base \( b \) be raised to result in \( a \)?"
- For example, \( \log_4 64 \) is 3 because \( 4^3 = 64 \).
- **Logarithm Exponent Rule**: One important rule is \( \log_b b^x = x \), meaning if the base of the logarithm and the base of the exponentiation are the same, the logarithms "cancel out" exponentiation, leaving you with the exponent.
- Therefore, for \( \log_4 4^7 \), based on this property, the result is simply 7, illustrating the cancelling effect because the base in both cases is 4.
Base of Logarithm
In the context of logarithms, the base is pivotal, as it determines how the logarithm operates. Think of the base like the lens through which you "view" the numbers in exponentiation.
- **Consistency in Bases**: When taking a logarithm, the base should remain consistent for simplification. For instance, in \( \log_4 \), the base is 4, matching with the exponent's base for \( 4^{7} \). This allows us to apply certain properties, like the logarithm exponent rule, seamlessly.
- **Base Must Be Positive and Not 1**: The base \( b \) of a logarithm must always be positive and not equal to 1. A base of 1 poses a problem because \( 1^x \) remains 1 for any exponent \( x \), making a meaningful logarithm calculation impossible.
These elements highlight why understanding the base is crucial to mastering logarithms.
- **Consistency in Bases**: When taking a logarithm, the base should remain consistent for simplification. For instance, in \( \log_4 \), the base is 4, matching with the exponent's base for \( 4^{7} \). This allows us to apply certain properties, like the logarithm exponent rule, seamlessly.
- **Base Must Be Positive and Not 1**: The base \( b \) of a logarithm must always be positive and not equal to 1. A base of 1 poses a problem because \( 1^x \) remains 1 for any exponent \( x \), making a meaningful logarithm calculation impossible.
These elements highlight why understanding the base is crucial to mastering logarithms.
Other exercises in this chapter
Problem 20
Determine whether each function is one-to-one. $$ f(x)=|x| $$
View solution Problem 21
Let \(f(x)=3 x\) and \(g(x)=4 x .\) Find each function and give its domain. See Example 1. $$ f+g $$
View solution Problem 21
Determine whether each function is one-to-one. $$ f(x)=x^{4} $$
View solution Problem 22
In this Study Set, assume that all variables represent positive numbers and \(b \neq 1\). Evaluate each expression. See Example \(1 .\) $$ \ln e^{8} $$
View solution