In Exercises 21–42, evaluate each expression without using a calculator. log5 5^7

Verified step by step guidance

1. Recognize the expression \( \log_5 5^7 \) as a logarithmic function.
2. Recall the logarithmic identity: \( \log_b b^x = x \).
3. Identify the base \( b \) as 5 and the exponent \( x \) as 7 in the expression \( \log_5 5^7 \).
4. Apply the identity: \( \log_5 5^7 = 7 \).
5. Conclude that the expression evaluates to the exponent, which is 7.

Key Concepts

Logarithms

Logarithms are the inverse operations of exponentiation. The logarithm of a number is the exponent to which a base must be raised to produce that number. For example, in the expression log_b(a), b is the base, and a is the number for which we are finding the logarithm. Understanding logarithms is essential for simplifying expressions involving exponents.

Properties of Logarithms

Logarithms have several key properties that simplify calculations. One important property is that log_b(b^x) = x, which states that the logarithm of a base raised to an exponent equals the exponent itself. This property is crucial for evaluating logarithmic expressions quickly and accurately, especially when the base and the argument are related.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = b^x, where b is a positive constant. These functions grow rapidly and are fundamental in various fields, including finance and natural sciences. Understanding the relationship between exponential functions and logarithms helps in evaluating expressions like log_b(a) by recognizing how they relate to each other.