Problem 45 - College Algebra

Question

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

Step-by-Step Solution

Answer

The value of the expression is 4.

Step 1: Understanding the Logarithmic Expression

We need to evaluate the expression \(6 \log _{8}(4)\) without using a calculator. We will use logarithm properties and basic math calculations.

Step 2: Rewrite the Logarithm in a Different Base

Notice that \(4 = 2^2\), so we can use the change of base formula. Write \(4\) as \(2^2\): \[\log_{8}(4) = \log_{8}(2^2).\]

Step 3: Apply the Power Rule of Logarithms

Using the power rule \(\log_{b}(a^n) = n\cdot \log_{b}(a)\), we have:\[\log_{8}(2^2) = 2 \cdot \log_{8}(2).\]

Step 4: Evaluate \(\log_{8}(2)\) Using Base Conversion

Since \(8 = 2^3\), we can write \(\log_{8}(2)\) as \(\log_{2^3}(2)\). Using the properties of logarithms, we know\[\log_{b^n}(a) = \frac{1}{n} \cdot \log_{b}(a),\]so \[\log_{2^3}(2) = \frac{1}{3} \cdot \log_{2}(2) = \frac{1}{3} \cdot 1 = \frac{1}{3}.\]

Step 5: Calculate \(6 \log_{8}(4)\)

Now substitute back \(\log_{8}(4) = 2 \log_{8}(2) = 2 \times \frac{1}{3} = \frac{2}{3}\) into the original expression:\[6 \log_{8}(4) = 6 \times \frac{2}{3} = 4.\]

Key Concepts

Change of Base Formula

Logarithms can often seem complex at first, but they become much easier to manage when you understand the change of base formula. This formula is especially useful when you want to simplify a logarithmic expression or when you're working with bases that are not convenient for calculation by hand.
The change of base formula states that for any positive numbers \(a, b,\) and \(c\) where \(a eq 1\) and \(c eq 1\), you can convert a logarithm to a different base using the formula:

• \(\log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)}\)

This allows you to change the base of the logarithm to one that might be simpler to work with, such as base 10 or base 2. Many students find this formula particularly helpful when evaluating logarithms without a calculator, as certain bases lend themselves to simpler calculations.

Power Rule of Logarithms

The power rule of logarithms is a handy tool when dealing with expressions that involve exponents inside a logarithm. This rule states that if you have a logarithm of a number raised to a power, you can bring the power out in front of the logarithm. Specifically, for any positive number \(a\) and any real number \(n\), the power rule is written as:

• \(\log_{b}(a^n) = n \cdot \log_{b}(a)\)

This can significantly simplify your calculations. Instead of dealing with a complex power inside a logarithm, you transform the problem into a multiplication, which is often easier to handle. For example, if you have \(\log_{8}(2^2)\), you can simply write it as 2 \(\cdot \log_{8}(2)\). This approach cuts down your work significantly and makes it easier to see the relationship between the numbers involved.
It is essential to remember this rule, as it can take what appears to be an intimidating expression and simplify it into a straightforward multiplication problem.

Logarithm Properties

The properties of logarithms are foundational tools for manipulating and simplifying logarithmic expressions. Frequently, these properties allow us to turn a complex problem into a more manageable form. Understanding these properties can help you solve problems more efficiently and effectively.
Here are a few fundamental properties of logarithms every student should know:

• Product Rule: \(\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)\). This helps in breaking down multiplication inside a logarithm into simple addition.
• Quotient Rule: \(\log_{b}(\frac{x}{y}) = \log_{b}(x) - \log_{b}(y)\). This rule is useful for converting division inside a logarithm to subtraction.
• Power Rule: As discussed earlier, \(\log_{b}(a^n) = n \cdot \log_{b}(a)\). It simplifies logarithms of powers by converting them to multiplications.
• Base Change Rule: This helps convert logarithms to simpler bases, especially when computation needs to be simplified.

By mastering these properties, you can often solve logarithmic expressions confidently, even without a calculator. They play an integral role in reducing the stress of dealing with complex logarithmic operations by providing clear strategies for simplification.