Problem 42 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$\log _{1 / 3}\left(\frac{1}{27}\right)$$

Step-by-Step Solution

Answer

3

Step 1: Identify the base of the logarithm

The logarithmic expression is given as \( \log_{1/3}\left(\frac{1}{27}\right) \). The base here is \( \frac{1}{3} \).

Step 2: Express 1/27 with the same base

Since \( 27 = 3^3 \), we can rewrite \( \frac{1}{27} \) as \( 3^{-3} \). Thus, the expression becomes \( \log_{1/3}(3^{-3}) \).

Step 3: Apply the change of base to match the base of the logarithm

The base of the logarithm is \( \frac{1}{3} \), which can be rewritten as \( 3^{-1} \). Therefore, the expression is \( \log_{3^{-1}}(3^{-3}) \).

Step 4: Simplify the logarithm using the properties

Using the property \( \log_{a^b}(a^c) = \frac{c}{b} \), we can simplify \( \log_{3^{-1}}(3^{-3}) \) as \( \frac{-3}{-1} = 3 \).

Step 5: Conclusion

Thus, the simplified form of the expression \( \log _{1 / 3}\left(\frac{1}{27}\right) \) is 3.

Key Concepts

Change of Base Formula

The Change of Base Formula is a handy tool when dealing with logarithms of uncommon bases. This formula allows us to express a logarithm in terms of logs of any base we choose, usually 10 or e. This is particularly useful when calculators only support these bases. The Change of Base Formula is stated as follows:\[\log_b a = \frac{\log_c a}{\log_c b}\]where:

• \(b\) is the original base of the logarithm.
• \(a\) is the argument of the logarithm.
• \(c\) is the new base you are converting to.

In practice, the formula isn't just useful for numerical calculations. It is also instrumental in solving problems. For example, when encountering bases that can be complex to work with, like fractions, converting them to integer bases simplifies the process. Thus, the Change of Base Formula becomes a vital shortcut in many logarithmic calculations. Using it effectively can remove the barrier of unfamiliar bases and turn every problem into a manageable common base format. Remember to always practice applying this formula, as it helps build a strong foundation in logarithmic manipulation.

Logarithmic Properties

Understanding the properties of logarithms is crucial, as they simplify complex problems and reveal insightful relationships between mathematical elements. Here are some fundamental properties you should know:

• **Product Property:** \( \log_b (xy) = \log_b x + \log_b y \)
• **Quotient Property:** \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \)
• **Power Property:** \( \log_b (x^y) = y \cdot \log_b x \)
• **Change of Base Property:** As mentioned before.

These properties allow for a more straightforward manipulation of logarithmic expressions. For instance, they can be used to simplify expressions by breaking down larger problems into smaller, manageable pieces.In solving the exercise, one notable application was the Power Property, which states that the logarithm of a power can be expressed as the exponent times the logarithm of the base. In the simplification process of this exercise, this property helped transition from the complicated expression to a readable one by rewriting \( \log_{3^{-1}}(3^{-3}) \) as a simple fraction \( \frac{-3}{-1} \). Understanding and applying these properties enables effective problem-solving and builds confidence in working with logarithms.

Exponents and Logarithms

Exponents and logarithms have an interwoven relationship, acting as inverse functions of each other. Understanding this relationship is foundational in mathematics, especially in problem-solving involving exponential growth or decay. Here’s a brief overview:

• When \( b^c = a \), then \( \log_b a = c \).
• This means that a logarithm finds the exponent we need to raise the base to, in order to get the argument.

In simpler terms, if you know the exponential form, the logarithm tells you what power you raise the base to get the desired result. This interplay was clearly seen in the original exercise.For example, to simplify \( \log_{1/3}(1/27) \), it was first expressed in exponential terms as \( 3^{-3} \). Recognizing logarithmic relationships like this lets us connect the dots in solving real-world problems. Many students find the transition between exponents and logarithms challenging. However, focus on their inverse nature and practice with various bases and powers to solidify understanding. In time, these concepts become intuitive, allowing a seamless transition between them during the solution process.