Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite the expression as a single logarithm and simplify the result. (Hint: Begin by using the properties of logarithms.) $$\ln (1+\sin x)-\ln |\sec x|$$

Step-by-Step Solution

Answer

The simplified expression is \( \ln((1+\sin x)*\cos x) \)

Step 1: Apply the logarithm subtraction rule

Replace the given expression \( \ln (1+\sin x)-\ln |\sec x|\) with \( \ln \frac{1+\sin x}{|\sec x|}\) using the property of logarithms. Here, the property used is that the subtraction of two logarithms is equivalent to the logarithm of the quotient of their arguments.

Step 2: Simplify the expression

Before we simplify, it would be helpful to recall what secant of x, denoted as \( \sec x \), is equivalent to. \( \sec x \) is equivalent to \( \frac{1}{\cos x} \). Thus, the expression become \( \ln \frac{1+\sin x}{|\frac{1}{\cos x}|} \) which simplifies to \( \ln((1+\sin x)*|\cos x|)\).

Step 3: Handle the absolute value

Since \( |\cos x| \) is always nonnegative ( i.e., \( |\cos x| >= 0 \) ), and \( (1+\sin x) \) is always positive (to be exact \(1+\sin x\in(0, 2]) \), then we can say that the function \( (1+\sin x)*|\cos x| \) is always nonnegative. Therefore, we can safely drop the absolute value, so that our expression becomes \( \ln((1+\sin x)*\cos x) \).

Key Concepts

Properties of Logarithms

Logarithms have amazing properties that make them essential in simplifying expressions. When dealing with logarithmic expressions, understanding these properties can make your life much easier! One fundamental property is that the subtraction of two logarithms is equivalent to the logarithm of the division of their arguments. This is formally written as:

• \( \ln(a) - \ln(b) = \ln \left( \frac{a}{b} \right) \)

This property is essential for reducing complex expressions into simpler, more manageable forms. By recognizing when you can apply these rules, you can often transform a tough-looking problem into something that's quite straightforward to handle. Keep these properties in mind as they can not only help with solving problems but also provide deep insights into the behavior of logarithmic functions themselves.

Simplifying Logarithms

Simplifying logarithms can turn a complex expression into something much clearer. After applying the properties of logarithms, the next step is usually simplification.
This involves understanding the underlying components of the expression.In our example, the key to simplification lies in the trigonometric identities. Recognizing that \( \sec x \) is equivalent to \( \frac{1}{\cos x} \), we can replace \( \ln |\sec x| \) with \( \ln |\frac{1}{\cos x}| \).
This allows us to transform our expression to \( \ln \frac{1 + \sin x}{|\frac{1}{\cos x}|} \) which simplifies to \( \ln((1 + \sin x) \cdot |\cos x|) \).Understanding such transformations is crucial. It involves recognizing equivalent expressions and the ability to manipulate them using known identities to reach a simpler, unified form. This skill is central to working with logarithms effectively.

Trigonometric Functions

Trigonometric functions often appear in logarithmic expressions, and knowing how to handle them is vital. In this example, we encounter \( \sin x \) and \( \sec x \). A key identity to remember is that \( \sec x = \frac{1}{\cos x} \). Trigonometric identities like this enable us to substitute and simplify expressions involving trigonometric and logarithmic combinations.
When you see expressions like \( \ln(1 + \sin x) \) or \( \ln |\sec x| \), know that there's a wealth of trigonometric concepts to explore.

• Always consider rewriting trigonometric functions using their fundamental identities.
• Evaluate the domain and range to understand any constraints.

By mastering these trigonometric basics, you'll find that simplifying and understanding logarithmic expressions is much more intuitive, and applying identities becomes second nature.