Problem 18 - Precalculus

Question

Evaluate the expression. \(\log _{2}(32)\)

Step-by-Step Solution

Answer

\(\log _{2}(32) = 5\).

Step 1: Understand the Problem

We need to evaluate the logarithmic expression \( \log_{2}(32) \). This is asking us to find the power to which the base 2 must be raised to obtain 32.

Step 2: Express 32 as a Power of 2

Determine how many times you need to multiply 2 by itself to reach 32. Start by listing the powers of 2: - \( 2^1 = 2 \)- \( 2^2 = 4 \)- \( 2^3 = 8 \)- \( 2^4 = 16 \)- \( 2^5 = 32 \).We see that \(32 = 2^5\).

Step 3: Solve for the Logarithm

Since \(32 = 2^5\), by definition of a logarithm, \(\log_{2}(32) = 5\). This means 2 to the power of 5 equals 32.

Step 4: Verify the Solution

To verify, calculate \(2^5\) and check if it equals 32. \(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\). Thus, our solution is correct.

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of the same number by itself. For example, when we write \(2^5\), we are saying that the number 2 is multiplied by itself 5 times, which equals 32. Exponents are a shorthand way to simplify expressions and calculations by showing that a number is multiplied a certain number of times. This concept is crucial not just in algebra, but in various domains of math.

• The base is the number that is being multiplied.
• The exponent tells us how many times the base is used as a factor.

Understanding how to manipulate exponents can help in transforming expressions, making calculations easier. For instance, knowing \(2^5 = 32\) allows us to quickly calculate the result without multiplying each time.

Exploring Base-2 Logarithms

A base-2 logarithm, denoted as \(\log_{2}(x)\), specifically refers to logarithms where the base number is 2. It answers the question: "To what power must 2 be raised, to equal x?" Let's take \(\log_{2}(32)\) for instance:

• This is asking for the power of 2 that results in 32.
• From our understanding of exponents, since \(2^5 = 32\), it tells us that \(\log_{2}(32) = 5\).

Using base-2 logarithms is particularly prevalent in computer science fields, where things are often measured in powers of 2 due to the binary nature of data. For example, understanding how data storage or processing power increases logarithmically allows insights into efficiency and capacity.

The Concept of Powers of 2

The phrase "powers of 2" refers to expressions like \(2^n\), where n is a whole number. Each power represents a doubling of the previous number. Starting from \(2^0 = 1\), every subsequent power increases exponentially:

• 2 raised to the power of 1 (\(2^1\)) equals 2.
• 2 raised to the power of 2 (\(2^2\)) equals 4.
• This sequence continues as 8, 16, 32, and so forth.

This concept is crucial as it underpins the binary system in computing. Since everything a computer processes is in binary form, understanding powers of 2 provides insight into memory allocation, machine learning algorithms, and network data flow. Knowing that \(32 = 2^5\) assists in comprehending the operation and scaling of computer-based logic and architecture.