Problem 26 - Algebra 2

Question

Use the Change of Base Formula to evaluate each expression. Then convert it to a logarithm in base \(8 .\) $$ \log _{4} 8 $$

Step-by-Step Solution

Answer

The expression \(\log_4(8)\) using the change of base formula and converted to base 8 is \(log_8(2)\).

Step 1: Analyze given expression

The given expression to be evaluated is \(\log_4(8)\). Hemce, the base is 4 and the argument is 8.

Step 2: Apply the Change of Base Formula

Using the change of base formula, let the new base to be 2, a common factor of both 4 and 8. This formula translates \(\log_b(a)\) to \(\frac{log_c(a)}{log_c(b)}\), so \(\log_4(8)\) becomes \(\frac{\log_2(8)}{\log_2(4)}\).

Step 3: Find the Result

By the properties of logarithms, \(\log_2(8) = 3\) and \(\log_2(4) = 2\). So \(\frac{\log_2(8)}{\log_2(4)} = \frac{3}{2}\).

Step 4: Convert to base 8

Now, the final step is to convert the resulting base 2 logarithm to base 8. Because \(8 = 2^3\), \(\frac{3}{2}\) can be expressed as \(3 * (\frac{1}{2})\). With the logarithm rule \(a*log_b(c) = log_b(c^a)\), \(\frac{3}{2}\) in base 2 logarithm can be changed to base 8 with the value equal to \(log_8(2)\).

Key Concepts

Understanding Logarithms

Logarithms are mathematical operations that help us determine the power to which a number, called the base, must be raised to produce another given number. For example, if we have \( \log_b(x) \), \( b \) is the base, and \( x \) is the number resulting from multiplying the base by itself a certain number of times.
Logarithms are the inverse operations of exponentiation, just like subtraction is the inverse of addition. They play a key role in simplifying complex calculations and are commonly used in various fields, including science and engineering. When using logarithms, it's crucial to understand the concept of the base since changing the base can alter the complexity and ease of calculation.
With the basics of logarithms in mind, you can better appreciate how they can convert multiplicative processes into additive ones, making them extremely useful in breaking down complex problems.

Base Conversion with Logarithms

The Change of Base Formula is a powerful tool when working with logarithms, especially when your calculator or computational tool does not directly support a particular base. This formula allows you to convert logarithms from one base to another.
The formula is:

• \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \)

where \( c \) is the new base of your choice.
This approach is useful because most calculators have \( \log_{10} (\text{logarithm base 10}) \) and \( \ln (\text{natural logarithm, base \( e \)}) \) functions built-in. By choosing \( c \) as either 10 or \( e \), you can easily calculate \( \log_b(a) \) using these functions.
In our exercise, we changed the base from 4 to 2, a common factor of both 4 and 8. This simplification made it easier to evaluate \( \log_4(8) \). This step is fundamental because it shows that any logarithm can be evaluated if you understand how to transform bases effectively.

Properties of Logarithms

Logarithms have a set of properties that make them highly useful and efficient when solving mathematical problems. One key property is the power rule, which is used in our solution for simplifying results. This rule states:

• \( \log_b(a^c) = c \cdot \log_b(a) \)

allowing you to take an exponent outside the logarithm.
Another important property is the product rule:

• \( \log_b(xy) = \log_b(x) + \log_b(y) \)

and the quotient rule:

• \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)

In our exercise, the property that \( \log_2(8) = 3 \) arises because 8 is \( 2^3 \), highlighting the usefulness of these properties in simplifying logarithmic tasks.
Understanding and remembering these properties allow you to manipulate logarithms effectively and thus solve complex problems with greater ease and precision.