Problem 45 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the base \(b\) logarithmic expression without using a calculator. \(6 \log _{8}(4)\)

Step-by-Step Solution

Answer

4

Step 1: Understand Logarithm Properties

We need to simplify the expression \(6 \log_{8}(4)\). Recall that the logarithmic property states \(a \log_{b}(c) = \log_{b}(c^a)\). We'll use this to simplify the expression.

Step 2: Apply Logarithm Power Rule

Apply the power rule to the expression: \(6 \log_{8}(4) = \log_{8}(4^6)\). This means we will raise 4 to the power of 6.

Step 3: Compute the Exponent

Calculate \(4^6\). First, find \(4^3\), which is \(64\). Then multiply \(64 \times 64\) to get \(4096\). Thus, \(4^6 = 4096\).

Step 4: Simplify the Logarithm

Substitute \(4^6 = 4096\) back into the expression: \(\log_{8}(4096)\). Now simplify \(\log_{8}(4096)\) knowing that \(4096 = 8^4\).

Step 5: Final Calculation

Since \(4096 = 8^4\), it follows that \(\log_{8}(4096) = \log_{8}(8^4) = 4\) using the property \(\log_{b}(b^a) = a\). Hence, the final value is 4.

Key Concepts

Logarithm Properties

Logarithmic functions are unique mathematical tools that help us work with large numbers, exponents, and multiplication in a simplified way. By understanding logarithm properties, solving expressions like \(6 \log_{8}(4)\) becomes manageable.A primary property involves the conversion between multiplication and exponentiation. Specifically, the property \(a \log_{b}(c) = \log_{b}(c^a)\) helps us rewrite products as powers. This rule is essential because it enables us to handle complex multiplications by expressing them as a single base raised to a power. For example, in our expression \(6 \log_{8}(4)\), we can transform it into \(\log_{8}(4^6)\). This step often simplifies calculations and is foundational to manipulating logarithms.Another important property is that \(\log_{b}(b) = 1\), meaning when the base and the argument are the same, the log value is always 1. Additionally, \(\log_{b}(1) = 0\) is a crucial property because any number raised to the power of zero results in 1. These simple yet powerful rules form the backbone of more intricate calculations with logarithms.

Logarithm Power Rule

The logarithm power rule allows us to handle expressions more efficiently. When using the power rule, you're effectively "pulling down" an exponent, turning a multiplication problem into something more akin to a single operation. In the expression \(6 \log_{8}(4)\), the coefficient 6 can be turned into the exponent of 4, creating \(\log_{8}(4^6)\).This transformation uses the formula \(a \log_{b}(c) = \log_{b}(c^a)\), enabling us to treat a complex expression as a straightforward power. Once in this format, the cumbersome multiplication is replaced by a single exponent. This approach harnesses the power rule to reduce theoretical complexity and aligns with techniques of simplifying exponential expressions. In practice, this makes solving real-world problems involving exponential growth or decay more approachable.Remember, knowing when and how to apply this rule can streamline calculations significantly, especially when manually solving logarithmic expressions without a calculator.

Exponent Calculation

Exponent calculation is a central step in solving problems involving logarithmic expressions. When applying the power rule in our example, we need to calculate \(4^6\). Understanding exponents involves multiplying a number by itself a specific number of times.To simplify the calculation, start by breaking it down: find \(4^3\) first. By multiplying 4 by itself three times, you get 64. Then, calculate \(64 \times 64\), which equals 4096. This means that \(4^6 = 4096\).In cases when calculators are off-limits, performing such calculations manually demands practice and familiarity with patterns in numbers. For instance:

• \(a^3 = a \times a \times a\)
• \(4^3 = 4 \times 4 \times 4 = 64\)
• Then, \(64 \times 64 = 4096\), giving \(4^6 = 4096\)

These steps illustrate how exponents work and why they are significant in simplifying logarithmic problems. By using basic multiplication techniques and understanding exponent rules, complex problems become more approachable and easier to solve.