Problem 40
Question
Write logarithm as a difference. Then simplify, if possible. \(\ln \frac{27}{e}\)
Step-by-Step Solution
Verified Answer
\(\ln \frac{27}{e} = \ln 27 - 1\).
1Step 1: Introduce the Logarithm Property
The expression \( \ln \frac{27}{e} \) can be rewritten using the logarithm property for quotients: that is, \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \).
2Step 2: Apply the Property
Using the formula from Step 1, the expression becomes \( \ln 27 - \ln e \).
3Step 3: Simplify the Expression
Knowing that \( \ln e = 1 \) (since the natural logarithm of \( e \) is 1), we can simplify the expression to \( \ln 27 - 1 \). This is the simplified form of the logarithmic expression written as a difference.
Key Concepts
Understanding Logarithmic PropertiesThe Natural Logarithm ExplainedProcess of Simplifying Expressions
Understanding Logarithmic Properties
Logarithmic properties are essential tools that help us manipulate and simplify logarithmic expressions. One important property is the quotient rule, which states that
Besides the quotient rule, other logarithmic properties include the product rule \( \ln(ab) = \ln a + \ln b \) and the power rule \( \ln(a^b) = b \ln a \).
These properties allow for flexibility in expressing logarithms, making it easier to solve equations. They're based on fundamental principles of logarithms and are key in transitioning between different types of mathematical problems.
- \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \)
Besides the quotient rule, other logarithmic properties include the product rule \( \ln(ab) = \ln a + \ln b \) and the power rule \( \ln(a^b) = b \ln a \).
These properties allow for flexibility in expressing logarithms, making it easier to solve equations. They're based on fundamental principles of logarithms and are key in transitioning between different types of mathematical problems.
The Natural Logarithm Explained
The natural logarithm, denoted as \( \ln \), is a specific type of logarithm that uses the irrational number \( e \) as its base. The number \( e \) is approximately 2.71828 and is used frequently in natural growth processes and calculus. In expressions like \( \ln \left( \frac{27}{e} \right) \), the base \( e \) remains implicit, highlighting its foundational role in natural logarithms.
One crucial aspect of the natural logarithm is understanding that \( \ln e = 1 \). This is because \( e^1 = e \), leading directly to this simple yet important result. Recognizing this relationship helps streamline calculations and simplifications involving natural logarithms.
One crucial aspect of the natural logarithm is understanding that \( \ln e = 1 \). This is because \( e^1 = e \), leading directly to this simple yet important result. Recognizing this relationship helps streamline calculations and simplifications involving natural logarithms.
Process of Simplifying Expressions
Simplifying logarithmic expressions often involves applying properties of logarithms to rewrite and reduce the given expression. For the expression \( \ln \frac{27}{e} \), the process started with recognizing the quotient form, allowing us to apply the quotient rule.
The expression was initially rewritten as \( \ln 27 - \ln e \). Knowing that \( \ln e = 1 \), we substituted this value into the expression, resulting in \( \ln 27 - 1 \). This final, simpler expression is more concise and easier to understand or further manipulate.
Such simplification steps are crucial in mathematics because they make expressions more manageable and often reveal essential numerical relationships.
The expression was initially rewritten as \( \ln 27 - \ln e \). Knowing that \( \ln e = 1 \), we substituted this value into the expression, resulting in \( \ln 27 - 1 \). This final, simpler expression is more concise and easier to understand or further manipulate.
Such simplification steps are crucial in mathematics because they make expressions more manageable and often reveal essential numerical relationships.
- Simplifying helps remove extraneous complexity.
- It clarifies the mathematical ideas involved.
Other exercises in this chapter
Problem 40
Write each exponential equation as a logarithmic equation. See Example 2. $$ 10^{3}=1,000 $$
View solution Problem 40
Each of the following functions is one-to-one. Find the inverse of each function and express it using \(f^{-1}(x)\) notation. \(f(x)=\frac{2 x+6}{3}\)
View solution Problem 41
Solve each equation. See Example \(6 .\) $$ \log 2 x=4 $$
View solution Problem 41
Let \(f(x)=2 x+1\) and \(g(x)=x^{2}-1 .\) Find each of the following. $$ (f \circ g)\left(\frac{1}{2}\right) $$
View solution