Problem 258 - Calculus Volume 1

Question

Write the equation in equivalent logarithmic form. \(\left(\frac{1}{3}\right)^{3}=\frac{1}{27}\)

Step-by-Step Solution

Answer

The equivalent logarithmic form is \(\log_{\frac{1}{3}} \frac{1}{27} = 3\).

Step 1: Understand the Exponential Form

The given equation is in the exponential form: \(\left(\frac{1}{3}\right)^{3}=\frac{1}{27}\). Here, the base is \(\frac{1}{3}\), the exponent is 3, and the result is \(\frac{1}{27}\).

Step 2: Convert to Logarithmic Form

To convert an equation from exponential form \(b^c = a\) to logarithmic form, we use the equivalent form \(\log_b a = c\). In this case, \(b = \frac{1}{3}\), \(a = \frac{1}{27}\), and \(c = 3\).

Step 3: Write the Logarithmic Equation

By substituting the values into the logarithmic form, the equation becomes \(\log_{\frac{1}{3}} \frac{1}{27} = 3\).

Key Concepts

Exponential Form

In mathematics, the exponential form is a way to express repeated multiplication of a number. The equation provided, \(\left(\frac{1}{3}\right)^{3}=\frac{1}{27}\), is an example of an exponential expression. In this context, three key components define the exponential form:

• Base: This is the number that is multiplied by itself. In the example, the base is \(\frac{1}{3}\).
• Exponent: This tells us how many times the base is multiplied by itself. Here, the exponent is 3.
• Result (or power): This is the outcome of taking the base to the power of the exponent, which in this equation results in \(\frac{1}{27}\).

If we break it down, multiplying \(\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3}\) yields \(\frac{1}{27}\). Understanding these fundamental components makes converting between exponential and logarithmic forms intuitive.

Logarithmic Conversion

Logarithmic conversion involves rewriting an exponential equation in a different form called a logarithm. This process is crucial in algebra for solving problems involving exponential growth or decay. The exponential equation \(b^c = a\) can be written in logarithmic form as \(\log_b a = c\).
Here’s how the conversion works based on the example \(\left(\frac{1}{3}\right)^{3}=\frac{1}{27}\):

• Identify the Base: The same base \(\frac{1}{3}\) is used in the logarithmic form.
• Result of the Exponential Expression: \(\frac{1}{27}\) becomes the number we are taking the logarithm of, known as \(a\) in \(\log_b a = c\).
• Exponent as the Logarithm Result: The original exponent, 3, becomes the solution or the logarithm's result.

Thus, the logarithmic form of the equation is \(\log_{\frac{1}{3}} \frac{1}{27} = 3\). Converting back and forth helps in cases where solving the equation is simpler in a different form.

Mathematics Education

Mathematics education often uses concepts like exponents and logarithms as building blocks for more advanced topics. Understanding these concepts is crucial for students as they facilitate problem-solving and enhance mathematical literacy.
Here are a few tips to strengthen comprehension and ensure effective learning:

• Relate to Real-life Examples: Demonstrate how these mathematical concepts appear in real-world scenarios such as calculating compound interest or understanding sound intensity levels.
• Use Visual Learning: Diagrams and visuals can help illustrate how bases and exponents interact, aiding significantly in converting between exponential and logarithmic forms.
• Link to Prior Knowledge: Encourage students to relate new concepts to previous topics they have mastered, such as repeated addition and multiplication.
• Interactive Practice: Leveraging online tools or math games can make learning more engaging and solidify these foundational principles.

Focused effort on these strategies significantly improves the understanding and application of mathematical concepts like logarithms and exponents.