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

Verified step by step guidance

1. 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 \(\log_{10} 6 = \log_{10} (2 \times 3)\).
2. Use the logarithm product rule, which states that \(\log_b (xy) = \log_b x + \log_b y\), to rewrite the expression as \(\log_{10} 6 = \log_{10} 2 + \log_{10} 3\).
3. Substitute the given approximate values into the expression: \(\log_{10} 2 \approx 0.3010\) and \(\log_{10} 3 \approx 0.4771\).
4. Add the two logarithm values together: \$0.3010 + 0.4771$.
5. The result of this addition will give you the approximate value of \(\log_{10} 6\) without using a calculator.

Key Concepts

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.