Find each value. If applicable, give an approximation to four decimal places. log 0.0022

Verified step by step guidance

1. Recognize that the problem asks for \( \log 0.0022 \), which means the logarithm base 10 of 0.0022.
2. Rewrite 0.0022 in scientific notation to make the logarithm easier to handle: \( 0.0022 = 2.2 \times 10^{-3} \).
3. Use the logarithm property \( \log(ab) = \log a + \log b \) to separate the expression: \( \log(2.2 \times 10^{-3}) = \log 2.2 + \log 10^{-3} \).
4. Recall that \( \log 10^{-3} = -3 \) because \( \log 10^k = k \) for any integer \( k \).
5. Calculate \( \log 2.2 \) using a calculator or logarithm table, then add it to \( -3 \) to find the final logarithm value. If needed, approximate the result to four decimal places.

Key Concepts

Definition of Logarithms

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log base 10 of 100 is 2 because 10 squared equals 100. Understanding this helps in converting between exponential and logarithmic forms.

Properties of Logarithms

Logarithms have key properties such as the product, quotient, and power rules that simplify calculations. For instance, log(ab) = log a + log b and log(a^n) = n log a. These properties are useful for breaking down complex logarithmic expressions.

Evaluating Logarithms of Small Numbers and Approximations

When the argument of a logarithm is a small decimal (less than 1), the logarithm value is negative. Calculators or logarithm tables can approximate these values, often rounded to a specified number of decimal places, such as four decimals in this problem.