In Exercises 21–42, evaluate each expression without using a calculator. log4 1

Verified step by step guidance

1. Understand that ${\log }_{b}a$ represents the power to which the base $b$ must be raised to obtain $a$.
2. In this problem, we have ${\log }_{4}1$.
3. Recall that any number raised to the power of 0 is 1, i.e., $b^0=1$ for any base $b$.
4. Therefore, ${\log }_{4}1=0$ because 4 raised to the power of 0 equals 1.
5. Conclude that the expression evaluates to 0.

Key Concepts

Logarithms

A logarithm is the inverse operation to exponentiation, answering the question: to what exponent must a base be raised to produce a given number? For example, in the expression log_b(a), b is the base, and a is the number. Understanding logarithms is essential for evaluating expressions like log4 1.

Properties of Logarithms

Logarithms have several key properties that simplify calculations. One important property is that log_b(1) equals 0 for any base b greater than 0. This is because any number raised to the power of 0 equals 1. Recognizing this property is crucial for solving the expression log4 1.

Base of a Logarithm

The base of a logarithm indicates the number that is raised to a power. In the expression log4 1, the base is 4. Understanding the significance of the base helps in interpreting logarithmic expressions and applying the correct properties to evaluate them.