Given that log10 2 ≈ 0.3010 and log10 3 ≈ 0.4771, find each logarithm without using a calculator. log10 6

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

문제 93

5장, 문제 93

Given that log₁₀ 2 ≈ 0.3010 and log₁₀ 3 ≈ 0.4771, find each logarithm without using a calculator. log₁₀ 6

검증된 단계별 안내

Recognize that the logarithm of 6 can be expressed in terms of the logarithms of its prime factors. Since 6 = 2 × 3, write the expression as \(\log_{10} 6 = \log_{10} (2 \times 3)\).

Use the logarithm product rule, which states that \(\log_b (xy) = \log_b x + \log_b y\), to rewrite the expression as \(\log_{10} 6 = \log_{10} 2 + \log_{10} 3\).

Substitute the given approximate values into the expression: \(\log_{10} 2 \approx 0.3010\) and \(\log_{10} 3 \approx 0.4771\).

Add the two logarithm values together: \(0.3010 + 0.4771\).

The result of this addition will give you the approximate value of \(\log_{10} 6\) without using 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

Logarithms have specific properties that simplify calculations, such as the product rule: log_b(mn) = log_b(m) + log_b(n). This allows breaking down complex logarithms into sums of simpler ones, which is essential for solving log_10 6 using known values of log_10 2 and log_10 3.

Common Logarithms (Base 10)

Common logarithms use base 10 and are often denoted as log or log_10. Understanding that log_10 10 = 1 and how to interpret logarithms in base 10 helps in applying given approximate values like log_10 2 and log_10 3 to find other logarithms.

Approximation and Estimation

Using given approximate values of logarithms allows estimation without a calculator. This skill involves substituting known logarithm values into formulas and performing arithmetic operations to find the desired logarithm accurately.