Find each value. If applicable, give an approximation to four decimal places. See Example 1. log 10^12

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. Understanding logarithms is essential for solving equations involving exponential growth or decay.

Properties of Logarithms

Logarithms have several key properties that simplify calculations, such as the product, quotient, and power rules. For example, log_b(mn) = log_b(m) + log_b(n) and log_b(m/n) = log_b(m) - log_b(n). These properties allow for the manipulation of logarithmic expressions to solve complex problems.

Common Logarithm

The common logarithm is a logarithm with base 10, denoted as log(x) or log_10(x). It is widely used in scientific calculations and can be easily computed using calculators. For instance, log(10^12) simplifies to 12, as 10 raised to the power of 12 equals 10^12.