Problem 149 - College Algebra

Question

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

The evaluation of \(\log _{3} 81\) and \(2 \log _{3} 9\) is 4. From this, it can be concluded that \(\log_{3} 81\), \(2 \log_{3} 9\) and \(\log_{3} 9^{2}\) are all equal.

Step 1: Evaluation of \(\log _{3} 81\)

To evaluate \(\log_{3} 81\), think about it as a question: '3 to the power of what number gives 81?'. We know that \(3^{4}=81\). Therefore, \(\log_{3} 81 = 4.\)

Step 2: Evaluation of \(2 \log _{3} 9\)

This can be simplified using power rule of logarithm which states that \(a \cdot \log_{b} c = \log_{b} c^{a}\). Thus, \(2 \log_{3} 9 = \log_{3} 9^{2} = \log_{3} 81\). Following from step 1, we know that \(\log_{3} 81 = 4\). Therefore, \(2 \log_{3} 9 = 4\).

Step 3: Conclusion about \(\log _{3} 81\), and \(\log _{3} 9^{2}\)

From step 1 and 2, it can be concluded that \(\log_{3} 81 = 2 \log_{3} 9 = \log_{3} 9^{2}\). This demonstrates the power rule of logarithms.

Key Concepts

Evaluating Logarithms

Understanding how to evaluate logarithms is essential when working with exponential expressions. A logarithm asks the question: What exponent do we need to raise a given base to in order to get a certain number? For instance, when facing \text{\text{ if you see \text{\text{\(\log _{3} 81\) is asking, 'What power of 3 gives us 81?' To solve this without a calculator, it's helpful to be familiar with powers of small numbers. Because \text{\text{\text{\(3^4 = 81\)}, we can deduce that \text{\text{\text{\(\log_{3} 81 = 4\). To build your skills in evaluating logarithms, remember that exponential and logarithmic functions are inverses. Using this understanding, you can often evaluate logarithms by finding the relevant exponential relationship, which is a method that doesn't require a calculator.

Power Rule of Logarithms

The power rule for logarithms is a vital property that allows you to manipulate expressions for easier evaluation. This rule states that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number, which mathematically is expressed as \text{\text{\text{\(a \cdot \log_{b} c = \log_{b} c^{a}\) . In our example, when we encounter \text{\text{\text{\(2 \log _{3} 9\)}, applying the power rule gives us \text{\text{\text{\(\log_{3} 9^{2}\), which simplifies to \text{\text{\text{\(\log_{3} 81\) because \text{\text{\text{\(9^2 = 81\)}. This effectively reduces the problem to the previously evaluated \text{\text{\text{\(\log_{3} 81 = 4\). Mastering this rule not only simplifies complex logarithmic expressions but also reinforces the interconnectedness of logarithms and exponents.

Logarithm Without Calculator

In many situations, evaluating a logarithm without a calculator is not only feasible but also encourages a stronger conceptual understanding. Begin by looking for exponent relationships that match the logarithm's base and number. For example, if we try to solve \text{\text{\text{\(\log_{3} 81\) or \text{\text{\text{\(\log_{3} 9^{2}\) without a calculator, recognizing that \text{\text{\text{\(3^4 = 81\)} and \text{\text{\text{\(3^2 = 9\)}, helps us evaluate these logarithms as 4 and 2 respectively. The key is to understand the relationship between the base and the argument of the logarithm, and to utilize the properties of logarithms like the power rule. Practice with different bases and numbers to become proficient in recognizing these patterns and solving logarithms without a calculator.