Given that log10 2 ≈ 0.3010 and log10 3 ≈ 0.4771, find each logarithm without using a calculator. log10 √30

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

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Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

문제 99

5장, 문제 99

Given that log₁₀ 2 ≈ 0.3010 and log₁₀ 3 ≈ 0.4771, find each logarithm without using a calculator. log₁₀ √30

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Recognize that the logarithm of a square root can be rewritten using the property \(\log_b \sqrt{x} = \log_b x^{1/2} = \frac{1}{2} \log_b x\).

Rewrite the given expression \(\log_{10} \sqrt{30}\) as \(\frac{1}{2} \log_{10} 30\).

Express 30 as a product of its prime factors: \(30 = 3 \times 10\) or more precisely \(30 = 3 \times 10 = 3 \times (2 \times 5)\), but since we only have logs for 2 and 3, use \(30 = 3 \times 10\) and then break down \(\log_{10} 30\) as \(\log_{10} (3 \times 10)\).

Use the logarithm product rule: \(\log_b (xy) = \log_b x + \log_b y\), so \(\log_{10} 30 = \log_{10} 3 + \log_{10} 10\).

Substitute the known values: \(\log_{10} 3 \approx 0.4771\) and \(\log_{10} 10 = 1\), then multiply the sum by \(\frac{1}{2}\) to find \(\log_{10} \sqrt{30}\).

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Properties of Logarithms

Logarithms have specific properties that simplify calculations, such as the product rule (log_b(xy) = log_b(x) + log_b(y)) and the power rule (log_b(x^r) = r * log_b(x)). These allow breaking down complex expressions into simpler parts.

Change of Base and Given Logarithm Values

Using given logarithm values like log_10(2) and log_10(3) helps compute other logarithms by expressing numbers in terms of these bases. This avoids calculator use by substituting known values into logarithmic expressions.

Square Root as an Exponent

The square root of a number can be expressed as that number raised to the 1/2 power. This allows applying the power rule of logarithms, converting log(√x) into (1/2) * log(x), simplifying the calculation.