Given that log10 2 ≈ 0.3010 and log10 3 ≈ 0.4771, find each logarithm without using a calculator. log10 9/4

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 97

5장, 문제 97

Given that log₁₀ 2 ≈ 0.3010 and log₁₀ 3 ≈ 0.4771, find each logarithm without using a calculator. log₁₀ 9/4

검증된 단계별 안내

Recognize that the logarithm of a quotient can be expressed as the difference of logarithms: \(\log_{10} \left( \frac{9}{4} \right) = \log_{10} 9 - \log_{10} 4\).

Express 9 and 4 in terms of their prime factors: \(9 = 3^2\) and \(4 = 2^2\).

Use the logarithm power rule to rewrite the logs: \(\log_{10} 9 = \log_{10} (3^2) = 2 \log_{10} 3\) and \(\log_{10} 4 = \log_{10} (2^2) = 2 \log_{10} 2\).

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

Simplify the expression by performing the multiplications and then subtracting to find the value of \(\log_{10} \left( \frac{9}{4} \right)\).

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(xy) = log_b x + log_b y), quotient rule (log_b(x/y) = log_b x - log_b y), and power rule (log_b(x^n) = n log_b x). These allow breaking down complex expressions into simpler parts.

Change of Base and Given Logarithm Values

Using known logarithm values, like log_10 2 ≈ 0.3010 and log_10 3 ≈ 0.4771, helps compute other logarithms by expressing numbers in terms of these bases. This avoids calculator use by rewriting numbers as products or quotients of known values.

Expressing Numbers as Products or Quotients of Prime Factors

To use given logarithm values effectively, numbers should be factored into primes or known bases. For example, 9/4 can be written as (3^2)/(2^2), enabling the use of logarithm properties and known values to find the logarithm of the entire expression.