Given that log￬10 2 ≈ 0.3010 and log￬10 3 ≈ 0.4771, find each logarithm without using a calculator. log￬10 6

Key Concepts

Properties of Logarithms

Logarithms have several key properties that simplify calculations. The product property states that log_b(mn) = log_b(m) + log_b(n), allowing us to break down logarithms of products into sums. Similarly, the quotient property states that log_b(m/n) = log_b(m) - log_b(n), and the power property states that log_b(m^k) = k * log_b(m). These properties are essential for manipulating logarithmic expressions.

Change of Base Formula

The change of base formula allows us to compute logarithms in different bases. It states that log_b(a) = log_k(a) / log_k(b) for any positive k. This is particularly useful when we have logarithm values in one base and need to find values in another base, as seen in the question where we use known logarithm values to find log_10(6).

Logarithm of a Product

To find log_10(6), we can express 6 as a product of its prime factors: 6 = 2 * 3. Using the properties of logarithms, we can then calculate log_10(6) as log_10(2) + log_10(3). By substituting the given approximate values for log_10(2) and log_10(3), we can easily find the logarithm of 6 without a calculator.