Problem 44 - Algebra 2

Question

Assume that \(\log 4 \approx 0.6021, \log 5 \approx 0.6990,\) and \(\log 6 \approx 0.7782 .\) Use the properties of logarithms to evaluate each expression. Do not use your calculator. \(\log 24\)

Step-by-Step Solution

Answer

\(\log 24 \approx 1.3803\)

Step 1: Use Properties of Logarithms to Break Down the Expression

The laws of logarithms state that \(\log ab = \log a + \log b\). Using this property, we can break down the expression \(\log 24\) into the sum of two logarithms whose values we already know. Here 24 can be represented as the product of 4 and 6. Thus, the expression becomes \(\log 24 = \log 4 + \log 6\).

Step 2: Substitute the Given Values

Now we can substitute the given logarithmic values into the equation. This gives us: \(\log 24 = 0.6021 + 0.7782\).

Step 3: Calculate the Final Result

Adding these two numbers together, we find that \(\log 24 \approx 1.3803\).

Key Concepts

Properties of Logarithms

Logarithms have unique properties that make them powerful tools in algebra. One of the key properties is the Product Rule. This rule states that the logarithm of a product is the sum of the logarithms of the individual factors. In mathematical terms, if you have two numbers, say \( a \) and \( b \), then \( \log(ab) = \log a + \log b \).

Understanding this property is crucial because it simplifies complex logarithmic expressions into sums of simpler logarithms. For example, if you know \( \log 4 \) and \( \log 6 \) individually, you can easily determine \( \log 24 \) without directly computing it from scratch. Instead, you use \( \log 24 = \log 4 + \log 6 \).

This property is very helpful in algebraic calculations, particularly when dealing with large numbers or expressions that are products. It allows you to evaluate expressions without a calculator by breaking them down into manageable parts.

Logarithmic Expressions

Logarithmic expressions are mathematical phrases that involve logarithms and operations such as addition, subtraction, multiplication, and division. These expressions can often be simplified or rewritten using the properties of logarithms to make them easier to work with.

In the context of the given exercise, the expression \( \log 24 \) is a perfect example of how logarithmic expressions can be simplified. By recognizing that 24 can be expressed as the product of 4 and 6, the expression \( \log 24 \) is rewritten as \( \log 4 + \log 6 \). This transformation uses the product rule, making it easier to evaluate the expression using known logarithmic values.

Students often encounter logarithmic expressions in algebra, and understanding how to manipulate and simplify them using logarithmic rules is essential. It helps in solving equations, evaluating expressions, and even in real-world applications where logarithms are used, such as in calculating growth rates and scales in sciences.

Algebraic Calculations

Algebraic calculations with logarithms involve using various algebraic rules and the properties of logarithms to find solutions to problems. These calculations often require you to manipulate and solve expressions within the realm of logarithmic and exponential functions.

In the exercise, we used a straightforward example of algebraic calculations involving logarithms. By employing the product rule for logarithms and inserting known values, we found that \( \log 24 = 0.6021 + 0.7782 \), which simplifies to approximately 1.3803 without using a calculator.

To carry out these calculations efficiently, it's important to:

• Recognize patterns and factor numbers when possible for simpler expressions.
• Be familiar with logarithmic properties such as the product, quotient, and power rules.
• Practice by solving different types of logarithmic expressions to build confidence and skill.

By mastering these techniques, students will enhance their problem-solving skills and ability to perform algebraic operations involving logarithms.