Problem 28 - Precalculus

Question

Evaluate the expression. \(\log _{4}(8)\)

Step-by-Step Solution

Answer

The value of \( \log _{4}(8) \) is \( \frac{3}{2} \).

Step 1: Understanding the Logarithmic Expression

We need to evaluate \( \log_{4}(8) \). This expression asks what power we should raise the base, 4, to get 8.

Step 2: Change of Base Formula

When computing logarithms, it's often useful to use the change of base formula: \( \log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)} \). We'll use base 2 because both 4 and 8 are powers of 2.

Step 3: Convert to Base 2

Convert the expression using base 2: \( \log_{4}(8) = \frac{\log_{2}(8)}{\log_{2}(4)} \).

Step 4: Evaluate \(\log_{2}(8)\) and \(\log_{2}(4)\)

Calculate \( \log_{2}(8) \): since \( 8 = 2^3 \), \( \log_{2}(8) = 3 \). Calculate \( \log_{2}(4) \): since \( 4 = 2^2 \), \( \log_{2}(4) = 2 \).

Step 5: Final Calculation

Substitute these values back into the converted expression: \( \log_{4}(8) = \frac{3}{2} \).

Key Concepts

Understanding the Change of Base Formula

The change of base formula is a very useful tool when dealing with logarithms. It allows us to rewrite a logarithm in a different base, which can make calculations much easier. The formula is given as: \( \log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)} \). This means you can convert a logarithm with base \( b \) into a fraction of two logarithms with a new base \( c \).

• This formula is particularly helpful when dealing with bases that are not common or easy to work with.
• By using a base that makes calculations simpler, such as base 2 or 10, you can evaluate the original expression more easily.

In the given problem, we used the change of base formula to convert \( \log_{4}(8) \) into \( \frac{\log_{2}(8)}{\log_{2}(4)} \) because both 4 and 8 are powers of 2.
This strategy simplifies the task significantly, as calculating powers of 2 is straightforward.

Breaking Down Logarithmic Expressions

Logarithmic expressions can often seem intimidating, but breaking them down makes them easier to understand. A logarithmic expression, such as \( \log_{b}(a) \), asks the question: "To what power must we raise the base \( b \) to obtain the number \( a \)?"
This means if you have \( \log_{b}(a) = x \), \( b^x = a \).

• Understanding that logarithms are the inverse operation of exponentiation is key. Just as subtraction is the inverse of addition, logarithms undo exponentiation.
• In our example, \( \log_{4}(8) \) asks, "What power do we raise 4 to, in order to get 8?"

By using properties of logarithms and transformations like the change of base formula, these questions become manageable operations. Breaking complex problems into simpler parts often involves transforming the base and using properties of exponents.

The Role of Exponents in Logarithms

Exponents play a central role in understanding and evaluating logarithms. When evaluating a logarithm, you are essentially finding an exponent. If \( b^x = a \), it follows that \( \log_{b}(a) = x \).

• A strong understanding of exponent rules simplifies the process of working with logarithms.
• In our exercise, understanding that \( 8 = 2^3 \) and \( 4 = 2^2 \) helped us evaluate the logarithm using base 2.

Using the rule that \( b^x = a \), to find \( \log_{b}(a) \), you essentially find which exponent \( x \) makes the equation true.
This relationship between exponents and logarithms turns complex expressions into manageable challenges. Whenever you see a logarithm, think of it in terms of its related exponent. This mental conversion makes logarithmic expressions less daunting to tackle.