Problem 59 - Algebra for College Students

Question

Explain how to determine \(\log _{4} 76\) without using Property 10.9.

Step-by-Step Solution

Answer

\( \log_{4} 76 \approx 3.123 \).

Step 1: Understanding the Logarithm Notation

The expression \( \log_{4} 76 \) is asking us to find the power to which the base 4 must be raised to yield 76.

Step 2: Change of Base Formula

The change of base formula allows us to express \( \log_{b} a \) as \( \frac{\log_{c} a}{\log_{c} b} \). We will use base 10 for our calculations. So, \( \log_{4} 76 = \frac{\log_{10} 76}{\log_{10} 4} \).

Step 3: Calculate Logarithms Using a Calculator

Using a calculator, find \( \log_{10} 76 \) and \( \log_{10} 4 \). For this purpose, \( \log_{10} 76 \approx 1.8808 \) and \( \log_{10} 4 \approx 0.6021 \).

Step 4: Division of Calculated Logarithms

Now, divide the values obtained: \( \frac{1.8808}{0.6021} \approx 3.123 \).

Step 5: Interpret the Result

The result, approximately 3.123, indicates that 4 raised to the power of 3.123 gives a number close to 76.

Key Concepts

Exponential Functions

Exponential functions play a fundamental role in mathematics. They have the form \( f(x) = a^x \), where \( a \) is a constant and \( x \) is the variable. When we encounter problems involving logarithms, understanding exponential functions becomes essential. It helps us grasp how logarithms "undo" the process of exponentiation.

Here’s the intuition behind it:

• Exponential function: For example, if \( a = 4 \), then \( 4^x \) is an exponential function where \( x \) is the exponent. If \( x = 3.123 \), then \( 4^{3.123} \approx 76 \).
• Inverse process: The logarithm serves as the opposite of an exponent. So, \( \log_4 76 \approx 3.123 \) means asking what power of 4 gives us 76.

By knowing this connection, understanding the logarithm becomes much simpler. It all boils down to figuring out the exponent in an exponential equation that results in a specific number.

Change of Base Formula

The change of base formula is a powerful tool for simplifying logarithmic calculations. It lets us convert logarithms from one base to another. This form is especially handy since calculators easily compute base 10 or natural logarithms.

The change of base formula is expressed as:

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

In the original problem, we needed to compute \( \log_4 76 \). Using the change of base formula, we transformed it into:

• \( \log_4 76 = \frac{\log_{10} 76}{\log_{10} 4} \)

This transformation makes it much easier to solve using a calculator. It allows us to deal with familiar base 10 logarithms. In practice, this method is essential because it aids in solving logarithmic problems whenever the calculator doesn’t support the base you need.

Logarithmic Calculations

Logarithmic calculations are the process of using logarithms to find unknown exponents. When we calculate logarithms like \( \log_{10} 76 \) and \( \log_{10} 4 \), we use calculators to obtain precise values.

In the exercise, these calculations were used for the division stage:

• Using a calculator: \( \log_{10} 76 \approx 1.8808 \)
• And for \( \log_{10} 4 \approx 0.6021 \)

Once these logarithmic values were obtained, they were divided:

• \( \frac{1.8808}{0.6021} \approx 3.123 \)

This division gives the exponent that answers the original question, revealing the power of 4 needed for 76. Mastering these calculations allows users to handle various logarithmic problems efficiently.