Problem 113 - Precalculus with Limits

Question

In Exercises 113 - 118, rewrite the expression as a single logarithm and simplify the result. \( \ln\mid\cos x\mid - \ln\mid\sin x\mid \)

Step-by-Step Solution

Answer

The expression \( \ln\mid\cos x\mid - \ln\mid\sin x\mid \) simplifies to \( \ln \mid\cot x\mid \).

Step 1: Apply the Rule of Logarithm Division

The rule of logarithm division states that the difference of two logarithms is equal to the logarithm of the quotient of their arguments. The difference given is \( \ln\mid\cos x\mid - \ln\mid\sin x\mid \). According to the rule, this can be rewritten as \( \ln \left( \frac{\mid\cos x\mid}{\mid\sin x\mid} \right) \).

Step 2: Simplify the Result

The expression inside the logarithm, \( \frac{\mid\cos x\mid}{\mid\sin x\mid} \), can be simplified as \( \mid\cot x\mid \), since cotangent is defined as the reciprocal of tangent and hence, \( \cot x = \frac{\cos x}{\sin x} \). Therefore, the expression simplifies to \( \ln \mid\cot x\mid \).

Key Concepts

Properties of Logarithms

Logarithms are an essential tool in mathematics for simplifying expressions and solving equations. Understanding their properties can greatly ease the manipulation of complex expressions.

• Product Rule: \( \log_b(MN) = \log_bM + \log_bN \). This property allows us to break down a logarithm of a product into a sum of logarithms.
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_bM - \log_bN \). This rule states that the logarithm of a quotient translates to the difference of two logarithms.
• Power Rule: \( \log_b(M^n) = n\log_bM \). With this rule, you can bring the exponent out in front as a multiplier.

For the given exercise, we used the Quotient Rule. The expression \( \ln\mid\cos x\mid - \ln\mid\sin x\mid \) uses this property to combine two logs into a single log: \( \ln \left(\frac{\mid\cos x\mid}{\mid\sin x\mid}\right) \).

Trigonometric Identities

Trigonometric identities are the backbone of trigonometry. They allow us to relate different trigonometric functions and simplify complex expressions.

• Reciprocal Identities: For instance, \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \). These identities relate functions like sine, cosine, and tangent to their reciprocals.
• Pythagorean Identities: These include \( \sin^2x + \cos^2x = 1 \). They're useful for transforming expressions involving squared trigonometric functions.
• Angle Sum and Difference Identities: They relate the sine and cosine of sums or differences of angles to products of sines and cosines of angles.

In the exercise, we applied the Reciprocal Identity to simplify the quotient expression \( \frac{\mid\cos x\mid}{\mid\sin x\mid} \) into \( \mid\cot x\mid \). This relies on knowing \( \cot x \) as the ratio of cosine to sine.

Simplifying Expressions

Simplifying expressions is a crucial step in problem-solving, which involves reducing a complex equation or expression to its simplest form. This process can make solving mathematical problems easier.

• Start by identifying common factors in expressions or terms that can cancel out.
• Use algebraic identities, like the distributive property, to combine or simplify terms.
• Recognize and apply appropriate rules of logarithms and trigonometry to transform the expression.

In the context of our exercise, simplifying involved applying both logarithmic and trigonometric identities to streamline the expression into \( \ln \mid\cot x\mid \). Understanding these steps allows us to see how powerful these mathematical tools can be when used correctly.