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

Verified step by step guidance

1. Recognize that the logarithm of a quotient can be expressed as the difference of logarithms: \(\log_{10} \left( \frac{9}{4} \right) = \log_{10} 9 - \log_{10} 4\).
2. Express 9 and 4 in terms of their prime factors: \$9 = 3^2\( and \)4 = 2^2$.
3. Use the logarithm power rule to rewrite the logs: \(\log_{10} 9 = \log_{10} (3^2) = 2 \log_{10} 3\) and \(\log_{10} 4 = \log_{10} (2^2) = 2 \log_{10} 2\).
4. 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$.
5. Simplify the expression by performing the multiplications and then subtracting to find the value of \(\log_{10} \left( \frac{9}{4} \right)\).

Key Concepts

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.