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

Verified step by step guidance

1. Recall the logarithm property for division: \(\log_{10} \left( \frac{a}{b} \right) = \log_{10} a - \log_{10} b\).
2. Apply this property to the given expression: \(\log_{10} \left( \frac{3}{2} \right) = \log_{10} 3 - \log_{10} 2\).
3. Substitute the given approximate values: \(\log_{10} 3 \approx 0.4771\) and \(\log_{10} 2 \approx 0.3010\).
4. Set up the subtraction: \$0.4771 - 0.3010$.
5. Perform the subtraction to find the approximate value of \(\log_{10} \left( \frac{3}{2} \right)\) (do this calculation yourself to complete the solution).

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify calculations, such as the quotient rule: log_b(a/c) = log_b(a) - log_b(c). This allows us to express the logarithm of a fraction as the difference of two logarithms, making it easier to compute values using known logs.

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 using given values like log_10 2 and log_10 3 helps in calculating other logarithms by applying logarithmic properties.

Using Given Logarithm Values

When specific logarithm values are provided, such as log_10 2 ≈ 0.3010 and log_10 3 ≈ 0.4771, these can be substituted directly into logarithmic expressions. This approach avoids calculator use and enables manual computation of related logarithms.