In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log₅ (7 × 3)

Verified step by step guidance

1. Recall the logarithmic property that states: \( \log_b (MN) = \log_b M + \log_b N \). This means the logarithm of a product can be expressed as the sum of the logarithms.
2. Apply this property to the given expression \( \log_5 (7 \times 3) \). Rewrite it as \( \log_5 7 + \log_5 3 \).
3. Check if either \( \log_5 7 \) or \( \log_5 3 \) can be simplified further. Since 7 and 3 are prime numbers and not powers of 5, these logarithms cannot be simplified further without a calculator.
4. Therefore, the expanded form of \( \log_5 (7 \times 3) \) is \( \log_5 7 + \log_5 3 \).
5. If needed, you can leave the expression in this expanded form as the final answer since no further simplification is possible without a calculator.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. The product rule states that log_b(M × N) = log_b(M) + log_b(N), which allows the expansion of logarithmic expressions involving multiplication into sums of logs.

Logarithmic Expansion

Logarithmic expansion involves rewriting a logarithmic expression as a sum, difference, or multiple of simpler logarithms using the properties of logarithms. This process simplifies complex expressions and can make evaluation easier.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator requires recognizing values that can be simplified using known logarithms or converting expressions to a form involving integers or simple fractions. For example, if the argument is a product of numbers with known logs, the expression can be expanded and simplified.