In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. ln x + ln 7

Verified step by step guidance

1. Recall the logarithmic property that states: \( \ln a + \ln b = \ln(ab) \). This property allows us to combine the sum of two logarithms into a single logarithm of the product of their arguments.
2. Identify the terms in the expression: \( \ln x + \ln 7 \). Here, \( a = x \) and \( b = 7 \).
3. Apply the property by multiplying the arguments inside the logarithms: \( \ln x + \ln 7 = \ln (x \times 7) \).
4. Simplify the product inside the logarithm: \( \ln (7x) \).
5. The expression is now condensed into a single logarithm with coefficient 1: \( \ln (7x) \).

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow combining or breaking down logarithmic expressions. For example, the product rule states that ln(a) + ln(b) = ln(ab), which is essential for condensing expressions.

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting multiple logarithms as a single logarithm. This involves applying logarithm properties to combine terms, ensuring the coefficient of the logarithm is 1, simplifying the expression for easier evaluation or further manipulation.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator involves recognizing values that simplify to known logarithmic results, such as ln(1) = 0 or ln(e) = 1. Using properties to rewrite expressions can help identify these values and simplify the evaluation process.