Problem 22 - Algebra 2

Question

Evaluate each logarithm. $$ \log _{2} 2^{5} $$

Step-by-Step Solution

Answer

5

Step 1: Understanding the properties of logarithms

Recall that for any base b and any positive integer n, \(\log_b b^n = n\). This means that the logarithm of b to the exponent n, with respect to base b, will always equal to the exponent itself n.

Step 2: Applying the properties of logarithms to our problem

Applying this property to our question, we see that we have a logarithm of base 2 of 2 to the power of 5. Octing the property, our result would be the exponent 5, because \(\log_2 2^5 = 5\).

Key Concepts

Properties of Logarithms

Logarithms are an essential part of mathematics because they help us solve equations involving exponential terms. One of the most important properties of logarithms is expressed as \( \log_b b^n = n \). This property states that if you have a logarithm with a base \(b\) and the same base raised to an exponent \(n\), the result is simply \(n\).

This is crucial because it simplifies the process of evaluating certain logarithms. For example, when you see \( \log_2 2^5 \), you can quickly recognize that the base of the logarithm (2) and the base of the exponent (also 2) are the same.

• This means, according to the property, the logarithm evaluates directly to the exponent, which is 5.

Understanding this property makes it easier to solve and analyze exponential functions. By recognizing the base-exponent relationship, you streamline calculations, making it faster and straightforward.

Exponents

Exponents are a mathematical way to represent repeated multiplication. When you see an expression like \(a^n\), it means that the base \(a\) is being multiplied by itself \(n\) times. This notation is not only a powerful way to handle large numbers but is also foundational in understanding logarithms.

Exponents follow certain rules:

• The product of powers rule: \(a^m \times a^n = a^{m+n}\).
• The power of a power rule: \((a^m)^n = a^{mn}\).
• The power of a product rule: \((ab)^n = a^n \times b^n\).

These rules show how to simplify complex expressions and solve exponential equations effectively. Recognizing these patterns helps in converting problems into solvable forms that can be handled using logarithms.

Evaluating Logarithms

Evaluating logarithms involves finding the exponent needed to raise a base to get a certain number. Sometimes it can seem complex, but by understanding the core concepts and properties, it becomes much clearer.

Consider the expression \(\log_b a\), which asks the question: "To what exponent must \(b\) be raised to equal \(a\)?" The answer to this question is the value of the logarithm.

• If you can express \(a\) as \(b^n\), then \(\log_b a = n\).

For example, using our exercise, \(\log_2 2^5\), we see that 2 raised to the power of 5 gives us the number we're interested in.

By applying the property of logarithms that connects exponents and logarithms, we're able to evaluate \(\log_2 2^5\) simply as 5. Practicing this method aids in building better intuition around logarithmic problems.