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)

검증된 단계별 안내

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.

Apply this property to the given expression \( \log_5 (7 \times 3) \). Rewrite it as \( \log_5 7 + \log_5 3 \).

Check if either \( \log_5 7 \) or \( \log_5 3 \) can be simplified further. Since 7 and 3 are not powers of 5, these logarithms cannot be simplified without a calculator.

Therefore, the expanded form of the expression is \( \log_5 7 + \log_5 3 \), which is the fully expanded form using properties of logarithms.

If needed, you can leave the answer in this expanded form or use a calculator to approximate the values, but the problem asks to expand without a calculator.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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), allowing the expansion of logarithmic expressions involving multiplication into sums of logs.

Change of Base and Evaluation

Evaluating logarithms without a calculator often involves expressing numbers as powers of the base or using known logarithmic values. Understanding how to rewrite expressions helps simplify or approximate values when direct calculation is not feasible.

Logarithmic Expression Expansion

Expanding logarithmic expressions means rewriting them using logarithm properties to break down complex arguments into simpler parts. This process aids in simplification, solving equations, or further manipulation in algebraic contexts.

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)