4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
a. ln secθ + ln cosθ

Verified step by step guidance

1. Recall the logarithm property that states: \(\ln a + \ln b = \ln (a \times b)\). This means the sum of two logarithms with the same base can be combined into the logarithm of the product of their arguments.
2. Identify the expressions inside the logarithms: here, we have \(\ln \sec \theta\) and \(\ln \cos \theta\).
3. Apply the property by multiplying the arguments inside the logarithms: \(\ln (\sec \theta \times \cos \theta)\).
4. Simplify the product inside the logarithm. Recall that \(\sec \theta = \frac{1}{\cos \theta}\), so multiply \(\sec \theta\) and \(\cos \theta\) accordingly.
5. Write the final expression as a single logarithm term: \(\ln (\sec \theta \cdot \cos \theta)\), which simplifies to \(\ln (1)\).

Key Concepts

Properties of Logarithms

Logarithmic properties include rules such as the product, quotient, and power rules. The product rule states that the sum of logarithms with the same base can be combined into the logarithm of the product of their arguments, i.e., ln(a) + ln(b) = ln(ab). This is essential for rewriting expressions as a single logarithmic term.

Trigonometric Functions and Identities

Understanding basic trigonometric functions like secant (sec θ = 1/cos θ) and cosine is crucial. Recognizing these relationships allows simplification of expressions inside logarithms, which helps in combining or reducing terms effectively.

Simplification of Logarithmic Expressions

Simplifying logarithmic expressions involves applying logarithm properties and algebraic manipulation to rewrite complex expressions into simpler or single terms. This skill is important for solving problems efficiently and for clearer interpretation of results.