Problem 75 - Intermediate Algebra

Question

Evaluate each logarithmic expression. See Examples 4 and 5. $$ \log _{2} 8 $$

Step-by-Step Solution

Answer

\( \log_{2} 8 = 3 \) because \( 2^3 = 8 \).

Step 1: Understand the Problem

We need to evaluate the logarithmic expression \( \log_{2} 8 \). This expression is asking, "To what power do we raise 2 to get 8?" To evaluate it, we need to convert it into an equation.

Step 2: Convert to Exponential Form

The logarithmic expression \( \log_{2} 8 = x \) can be rewritten in exponential form as \( 2^x = 8 \). This means we need to find the value of \( x \) such that when \( 2 \) is raised to the power of \( x \), it equals 8.

Step 3: Solve the Exponential Equation

The equation \( 2^x = 8 \) requires us to find \( x \). We know that \( 8 \) can be expressed as \( 2^3 \) (since \( 8 = 2 \times 2 \times 2 \)). Therefore, \( 2^3 = 8 \) implies that \( x = 3 \).

Step 4: Verification

Verify the solution by checking if \( 2^3 = 8 \) holds true. Since this statement is correct, the value of \( \log_{2} 8 \) is indeed 3.

Key Concepts

Exponential Form

Understanding the exponential form is crucial when dealing with logarithmic expressions. Logarithms and exponents are closely related. They are two sides of the same coin. In mathematics, an exponential form is a way of expressing numbers where a base is raised to a certain power (or exponent). For example, in the exponential expression \(2^x\), \(2\) is the base and \(x\) is the exponent. The exponential form tells us how many times to use the base in a multiplication.

Here's a simple breakdown:

• The base is the number you multiply.
• The exponent tells us how many times to multiply the base by itself.

To connect this with logarithms, converting a logarithmic expression to an exponential form is a common strategy to make solving easier. For instance, the logarithmic expression \(\log_{2} 8\) can be restated as \(2^x = 8\). This tells you that you need to figure out how many times you must multiply 2 to reach 8.

Solving Exponential Equations

Solving exponential equations is a necessary skill for evaluating logarithmic expressions. Once you've written an equation in exponential form, like \(2^x = 8\), your task is to find the value of the exponent \(x\).

Let's break down the process:

• Identify the base: Here, it's 2.
• Express the other number using the same base: Since 8 can be rewritten as \(2^3\), we know \(x\) must be 3.
• Set the exponents equal: \(x = 3\).

After solving, it’s always good to verify your solution. In our example, check that \(2^3 = 8\), confirming the accuracy of \(x = 3\). This verification step ensures that the calculated exponent makes sense and that no errors were made during the process.

Logarithms

Logarithms are the inverse functions of exponents. They help us determine the exponent that a given base must be raised to in order to produce a certain number. In essence, if \(b^x = y\), then \(\log_b y = x\). This reflects the direct relationship between logarithmic and exponential forms.

Consider the following key points about logarithms:

• The base of the logarithm and the base of the exponent are the same.
• The logarithm answers the question: "To what power must the base be raised to produce a certain number?"
• Logarithmic identities can simplify complex calculations, especially in scientific and engineering problems.

In the exercise at hand, \(\log_{2} 8\) asks us what exponent \(2\) must be raised to in order to produce 8. By solving using the corresponding exponential form, \(2^x = 8\), we determine that the answer is 3.