Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 0.0022

Verified step by step guidance

1. Step 1: Recognize that you need to find the logarithm of a number, specifically \( \log(0.0022) \).
2. Step 2: Understand that \( \log(x) \) is the power to which the base 10 must be raised to obtain the number \( x \).
3. Step 3: Rewrite 0.0022 in scientific notation. 0.0022 can be expressed as \( 2.2 \times 10^{-3} \).
4. Step 4: Use the property of logarithms that states \( \log(a \times b) = \log(a) + \log(b) \). Apply this to get \( \log(2.2 \times 10^{-3}) = \log(2.2) + \log(10^{-3}) \).
5. Step 5: Calculate \( \log(10^{-3}) \), which is \(-3\), and then find \( \log(2.2) \) using a calculator or logarithm table. Add these two results to find \( \log(0.0022) \).

Key Concepts

Logarithms

A logarithm is the power to which a base must be raised to produce a given number. In the expression log_b(a), 'b' is the base, 'a' is the number, and the result is the exponent 'x' such that b^x = a. Logarithms are essential for solving exponential equations and are widely used in various fields, including science and engineering.

Common Logarithm

The common logarithm is a logarithm with base 10, denoted as log(x) or log_10(x). It is frequently used in calculations involving large numbers and is particularly useful in scientific notation. For example, log(100) equals 2 because 10^2 = 100. Understanding common logarithms is crucial for interpreting and solving problems involving exponential growth or decay.

Negative Logarithm Values

Logarithms of numbers less than 1 yield negative values. For instance, log(0.0022) will result in a negative number because 0.0022 is less than 1. This property is important for understanding the behavior of logarithmic functions and their graphs, which show that as the input approaches zero from the right, the logarithm approaches negative infinity.