Problem 18 - Algebra 2

Question

Evaluate each logarithm. $$ \log _{2} 8 $$

Step-by-Step Solution

Answer

The value of \(\log_2 8\) is 3.

Step 1: Identify the base and the argument of the logarithm

In this exercise, the base 'b' of the logarithm is 2 and the argument 'a' is 8. This can be write as \(\log_2 8\). The goal is to find the exponent 'x' that makes \(2^x = 8\).

Step 2: Use the definition of the logarithm

The definition of the logarithm states that if \(\log_b a = x\), then \(b^x = a\). To use this definition in this exercise, we will set \(2^x = 8\) and solve for 'x'.

Step 3: Solve for 'x'

By observing the equation \(2^x = 8\), one can see that \(x = 3\) since \(2^3 = 8\).

Step 4: Verification

To validate the answer, replace value in the original equation. The equation becomes \(\log_2 8 = 3\) which is consistent with \(2^3 = 8\). This confirms that the solution 'x=3' is correct

Key Concepts

Understanding Base and Argument in Logarithms

Before diving into logarithms, it's important to understand the components involved. A logarithm such as \( \log_b a \) consists of two main parts:

• Base (\(b\)): This is the number that is repeatedly multiplied.
• Argument (\(a\)): This is the result or the number you aim to achieve by exponentiating the base.

In the exercise \( \log_2 8 \), the base is 2, and the argument is 8. This essentially asks us, "To what power must 2 be raised, to get 8?" Recognizing these components helps you set up the problem for further evaluation. It's crucial because it frames the way you will calculate or interpret the logarithm. You know you're solving for the exponent that links the base to the argument.

The Role of Exponentiation in Logarithms

Exponentiation is the process of raising a number (the base) to a particular power (the exponent). In the context of logarithms, exponentiation is the inverse operation that defines the relationship between the base and the argument. This means that if you know a base and want to determine the exponent that gives a specific number, you use a logarithm.

• For example, in \( 2^x = 8 \), exponentiation is used during the calculation to determine what power (\(x\)) results in 8.
• This is reversed to find the logarithm that represents this situation as \( \log_2 8 = 3 \).

Exponentiation underpins the functionality and usage of logarithms in various mathematical and practical applications. It's all about identifying how many times you multiply that base to reach the argument.
Understanding this connection is key to solving logarithms effectively.

Solving Logarithmic Equations

Solving logarithmic equations might sound tricky, but it's simply a series of logical steps. Here's how you can approach a problem like \( \log_2 8 \).

• Identify the Components: Recognize the base (2) and the argument (8).
• Use Logarithmic Definition: Applying the definition \( \log_b a = x \) means finding \( x \) such that \( b^x = a \).
• Set Up the Exponential Equation: In this case, you set \( 2^x = 8 \).
• Solve for the Exponent: By realizing that \( 2^3 = 8 \), you discover \( x = 3 \).

This solution proves that the equation is satisfied. Solving logarithmic equations involves translating between the logarithmic form and its equivalent exponential form. Once you grasp that, the task becomes manageable! These steps streamline understanding and make the process consistent each time.