In Exercises 81–100, evaluate or simplify each expression without using a calculator. log 10^7

Key Concepts

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in equations of the form b^y = x. The logarithm log_b(x) answers the question: to what power must the base b be raised to produce x? Understanding logarithms is essential for simplifying expressions involving powers and roots.

Properties of Logarithms

Logarithms have several key properties that simplify calculations. For instance, the power rule states that log_b(a^c) = c * log_b(a). This property allows us to bring exponents down in logarithmic expressions, making it easier to evaluate or simplify them without a calculator.

Common Logarithm

The common logarithm, denoted as log(x) or log_10(x), uses 10 as its base. It is frequently used in scientific calculations and simplifies many expressions involving powers of 10. Recognizing that log(10^7) can be simplified directly to 7 is crucial for quickly evaluating logarithmic expressions.