Problem 44 - Algebra 2 and Trigonometry

Question

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ \log _{2} 32 $$

Step-by-Step Solution

Answer

\( \log_{2} 32 = 5 \).

Step 1: Understanding Logarithms

The logarithm \( \log_b a \) asks the question: "To what power must we raise \( b \) to get \( a \)?" In this problem, we need to find \( x \) such that \( 2^x = 32 \).

Step 2: Converting to Exponential Form

We convert the expression \( \log_{2} 32 \) into its exponential form: \( 2^x = 32 \). This means we are looking for the power \( x \) that makes this equation true.

Step 3: Identify Power of 2

Recognize that 32 is a power of 2. In fact, 32 can be expressed as \( 2^5 \). This means that when we multiply 2 by itself 5 times, we get 32.

Step 4: Solve for the Logarithm

Since \( 2^5 = 32 \), it follows from the definition of a logarithm that \( \log_{2} 32 = 5 \). Therefore, the value of the logarithm is 5.

Key Concepts

Exponential Form

Exponential form is a way of expressing numbers using a base and an exponent. It is especially helpful when dealing with large numbers or powers. To understand it, think of it as a powerful form of multiplication where a number, known as the base, is multiplied by itself a certain number of times, which is the exponent. For instance, in the equation \( 2^x = 32 \), 2 is the base and \( x \) is the exponent. The exponential form clearly shows what exponent is needed to obtain the desired value.

• Base: The number that is multiplied by itself.
• Exponent: The number that tells us how many times to multiply the base by itself.

Converting a logarithmic expression into its exponential form can simplify solving mathematical problems. In our example, converting \( \log_2 32 \) to \( 2^x = 32 \) allows us to directly determine the exponent \( x \). Understanding exponential form is crucial when working with logarithms as it is key in rewiring the mathematical expressions into more manageable forms.

Power of 2

In mathematical expressions, particularly those involving exponential form or logarithms, understanding the concept of a power of 2 is crucial. A "power of 2" simply means multiplying 2 by itself a certain number of times. These sequences of numbers have a base of 2 being raised to a power.Let's break this down using our example with the number 32. When we look for what power of 2 equals 32, we write it as:

• \(2^1 = 2\)
• \(2^2 = 4\)
• \(2^3 = 8\)
• \(2^4 = 16\)
• \(2^5 = 32\)

This shows that multiplying the base number 2 by itself 5 times gives 32. Hence, 32 is \(2^5\).Recognizing that numbers can be expressed as powers of 2 is very helpful. Doing so can make complex problems much easier to solve, particularly when dealing with logarithms.

Logarithmic Expression

A logarithmic expression is a way of expressing an exponent indirectly. It shifts the focus from simple multiplication to understanding the power needed to attain a certain number. Logarithms answer the question, "To what power must the base be raised to achieve a specific number?" For instance, \(\log_2 32\) asks, "What power of 2 equals 32?"

• Logarithm Base: This tells us which number we are raising to a power. Here, the base is 2.
• Numerical Result: The number we want to reach by raising the base to the right power. In this case, it is 32.
• Exponent/Logarithm Result: The answer to the logarithmic expression; the power we raise the base by. It is 5 in our example.

Logarithmic expressions are invaluable for simplifying problems involving exponential growth or decline. They allow us to work backwards from a result to find what initially needed to be done, as shown in the exercise where \(\log_2 32 = 5\). Understanding this representation fosters a deeper comprehension of exponential relationships in mathematics.