Problem 152 - Algebra and Trigonometry

Question

Exercises 150–152 will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about $$ \log _{3} 81, \text { or } \log _{3} 9^{2} ? $$

Step-by-Step Solution

Answer

a. The result of \(\log _{3} 81\) is \(4\). b. The result of \(2 \log _{3} 9\) is also \(4\). When comparing \(\log _{3} 81\) and \(2 \log _{3} 9\), we understand that multiplying a constant to a logarithmic function is equivalent to raising the argument of the logarithm to the power of that constant.

Step 1: Logarithm Recognition

Firstly, recognize that \(81\) is \(3^4\) and \(9\) is \(3^2\). So the questions can be rewritten as \(\log _{3} 3^4\) and \(2 \log _{3} 3^2\).

Step 2: Evaluate Logarithm Expressions

Using the relationship that \(\log_b b^a = a\), the evaluations are straightforward. The first expression becomes \(4\) and the second one becomes \(2 * 2 = 4\).

Step 3: Comparison and Conclusion

Now compare \(\log _{3} 81\) and \(2 \log _{3} 9\), you see both evaluate to \(4\). Therefore, evaluating an expression where a number multiplies the log is equivalent to the number becoming the exponent inside the log, giving \(\log _{3} 9^2\). This verifies the Power Rule for logarithms.

Key Concepts

Logarithm Properties

Understanding the fundamental properties of logarithms is an essential step towards mastering their evaluation and manipulation. A logarithm, essentially, is the inverse operation to exponentiation. It answers the question of how many times we need to multiply a base number to obtain another number. The logarithmic function denoted as \( \text{log}_b(a) \), reads as 'log base b of a', and indicates how many times you need to multiply the base \( b \) to get the number \( a \).

There are three critical properties we need to know:

• The Product Rule tells us that \( \text{log}_b(mn) = \text{log}_b(m) + \text{log}_b(n) \), meaning the log of a product is equal to the sum of logs.
• The Quotient Rule states that \( \text{log}_b\frac{m}{n} = \text{log}_b(m) - \text{log}_b(n) \), so the log of a quotient is the difference of logs.
• The Power Rule says that \( \text{log}_b(m^n) = n \times \text{log}_b(m) \), which translates an exponent outside of the log into a multiplication.

In the exercise provided, recognizing how these properties can simplify logarithmic expressions was crucial. Through identifying that \( 81 = 3^4 \) and \( 9 = 3^2 \), we're able to reduce the complexity of the expressions using the direct relationship between logarithms and exponents.

Power Rule for Logarithms

The Power Rule for logarithms is a powerful tool that simplifies the process of working with logarithmic expressions involving exponents. As per this rule, for any positive real numbers \( a \), \( b \), and any real number \( n \), the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number: \[ \text{log}_b(a^n) = n \times \text{log}_b(a) \].

This property is beautifully illustrated in the given exercise where we see \( 2 \times \text{log}_3(9)\) simplified using the product of \( 2 \) and the logarithm of \( 9 \), which is basically \( 2 \times 2 \) due to \( 9 \) being \( 3^2 \). The result is the same as if we had directly applied the logarithm to \( 81 \), or \( 9^2 \), which strengthens our understanding of the power rule's function. It reinforces the idea that the exponent inside a logarithm can be brought out front, changing the order of operations without changing the value.

Operations with Logarithms

Performing operations with logarithms can often involve more than just applying properties; it requires a thoughtful approach to restructuring logarithmic expressions to make them simpler. When evaluating expressions like \( \text{log}_3(81) \) or \( 2 \times \text{log}_3(9) \), recognizing patterns and numbers as powers of the base of the logarithm simplifies the problem, as seen in the exercise.

Generally, when working with logarithms, it’s important to:

• Identify the base and rewrite complex numbers as powers of this base whenever possible.
• Apply logarithm properties such as the Product, Quotient, and Power Rules to break down and simplify expressions.
• Look for opportunities to use the fundamental definition of logarithms, which states that \( \text{log}_b(b^a) = a \), to directly find the values of logarithmic expressions.

The operations with logarithms can thus be seen as a puzzle, where recognizing the base and its powers allows one to apply the correct properties to evaluate the expression easily. In both educational and practical terms, mastering operations with logarithms offers a potent tool for solving exponential and logarithmic equations.