Problem 96
Question
Rewrite the expression as a single logarithm and simplify the result. (Hint: Begin by using the properties of logarithms.) $$\ln |\cot \theta|+\ln |\sin \theta|$$
Step-by-Step Solution
Verified Answer
The simplified single logarithm of the expression \( \ln |\cot \theta|+\ln |\sin \theta| \) is \( \ln |\cos \theta| \)
1Step 1: Understand the problem
The given expression is \( \ln |\cot \theta|+\ln |\sin \theta| \). The problem asks to rewrite the expression as a single logarithm.
2Step 2: Apply the logarithmic product rule
According to the logarithmic product rule the sum of two logarithms \( \ln a + \ln b \) equals \( \ln (a \cdot b) \). Therefore, the given expression can be rewritten as \( \ln (|\cot \theta| \cdot |\sin \theta|) \)
3Step 3: Simplify the expression
The expression \( |\cot \theta \cdot \sin \theta| \) simplifies to \( |\cos \theta| \). Therefore, \( \ln |\cot \theta|+\ln |\sin \theta| = \ln |\cos \theta| \)
Key Concepts
Logarithmic PropertiesTrigonometric FunctionsSimplifying Logarithms
Logarithmic Properties
Logarithmic properties are essential tools for simplifying and manipulating expressions involving logarithms. These properties include the rules for product, quotient, and power.
\(\)The product rule is particularly useful and states that the sum of two logarithms with the same base can be expressed as a single logarithm of the product of their arguments. Given an expression \( \ln a + \ln b \), using the product rule, we can rewrite it as \( \ln (a \cdot b) \).
This concept greatly simplifies solving logarithmic equations. When faced with compound logarithmic expressions, identifying opportunities to apply these rules can streamline the solution process. Essentially, they allow us to convert multiple logarithmic expressions into one, facilitating easier manipulation and understanding of the underlying relationships in the problem.
\(\)The product rule is particularly useful and states that the sum of two logarithms with the same base can be expressed as a single logarithm of the product of their arguments. Given an expression \( \ln a + \ln b \), using the product rule, we can rewrite it as \( \ln (a \cdot b) \).
This concept greatly simplifies solving logarithmic equations. When faced with compound logarithmic expressions, identifying opportunities to apply these rules can streamline the solution process. Essentially, they allow us to convert multiple logarithmic expressions into one, facilitating easier manipulation and understanding of the underlying relationships in the problem.
Trigonometric Functions
Trigonometric functions like sine, cosine, and cotangent are fundamental in mathematics. They relate the angles of a triangle to the lengths of its sides and have wide applications.
The cotangent, \( \cot \theta \), is the reciprocal of the tangent function and is defined as \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \). It plays a crucial role in the given problem where we have the expression \( \ln |\cot \theta| + \ln |\sin \theta| \). In this context, understanding how these functions work helps in simplifying the expression by transforming \( |\cot \theta| \cdot |\sin \theta| \) to \(|\cos \theta|\).
This conversion is possible because the \( \sin \theta \) terms cancel out, leaving \( |\cos \theta| \) as the product. Recognizing these relationships between trigonometric functions facilitates the process of simplification, turning complex expressions into more manageable forms.
The cotangent, \( \cot \theta \), is the reciprocal of the tangent function and is defined as \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \). It plays a crucial role in the given problem where we have the expression \( \ln |\cot \theta| + \ln |\sin \theta| \). In this context, understanding how these functions work helps in simplifying the expression by transforming \( |\cot \theta| \cdot |\sin \theta| \) to \(|\cos \theta|\).
This conversion is possible because the \( \sin \theta \) terms cancel out, leaving \( |\cos \theta| \) as the product. Recognizing these relationships between trigonometric functions facilitates the process of simplification, turning complex expressions into more manageable forms.
Simplifying Logarithms
Simplifying logarithms involves a series of steps that leverage specific mathematical properties to make the expression more direct or manageable.
In our original expression \( \ln |\cot \theta| + \ln |\sin \theta| \), we start by applying the product rule of logarithms, combining it into a single logarithmic expression. The expression then becomes \( \ln (|\cot \theta| \cdot |\sin \theta|) \).
This is where simplification techniques come into play. By understanding the relationships between trigonometric functions \( |\cot \theta| \cdot |\sin \theta| = |\cos \theta| \), we simplify the logarithmic expression further to \( \ln |\cos \theta| \). Many logarithmic problems can be demystified using these properties, primarily focusing on reducing the complexity in steps and ensuring that each transformation moves towards a clearer and more concise final result.
In our original expression \( \ln |\cot \theta| + \ln |\sin \theta| \), we start by applying the product rule of logarithms, combining it into a single logarithmic expression. The expression then becomes \( \ln (|\cot \theta| \cdot |\sin \theta|) \).
This is where simplification techniques come into play. By understanding the relationships between trigonometric functions \( |\cot \theta| \cdot |\sin \theta| = |\cos \theta| \), we simplify the logarithmic expression further to \( \ln |\cos \theta| \). Many logarithmic problems can be demystified using these properties, primarily focusing on reducing the complexity in steps and ensuring that each transformation moves towards a clearer and more concise final result.
Other exercises in this chapter
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