Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Use the definition of a logarithm to write the equation in exponential form. For example, the exponential form of \(\log _{5} 125=3\) is \(5^{3}=125\).\(\log _{27} 3=\frac{1}{3}\)

Step-by-Step Solution

Answer

The exponential form of the equation \(\log_{27} 3=\frac{1}{3}\) is \(27^{\frac{1}{3}}=3\).

Step 1: Identify the base, argument, and logarithm

From the logarithm equation \(\log_{27} 3=\frac{1}{3}\), the base \(b\) is 27, the argument \(a\) is 3, and the logarithm of a to base b (\(c\)) is \(\frac{1}{3}\).

Step 2: Translate to exponential form

Using the pattern of exponential form, \(b^{c} = a\), substituting the values gives us \(27^{\frac{1}{3}}=3\). So, the exponential form of \(\log_{27} 3 = \frac{1}{3}\) is \(27^{\frac{1}{3}}=3\).

Key Concepts

Exponential Form

Exponential form is a way of expressing a number using a base and an exponent. In mathematics, we use it to show repeated multiplication of a number. For instance, in the expression \(a = b^c\), \(b\) is the base, \(c\) is the exponent, and \(a\) is the result. This form is particularly useful for simplifying multiplication of the same number over and over.
A logarithm is inherently related to exponents. Specifically, if you know the logarithmic form, you can easily convert it to exponential form. For example, the equation \(\log_b a = c\) can be rewritten as \(b^c = a\). This transformation helps in solving various mathematical problems involving growth or decay, such as compound interest or radioactive decay. It’s crucial to practice changing between logarithmic and exponential form for fluency in recognizing and solving these equations.

Base and Argument Identification

Understanding logarithms begins with identifying the correct components: the base and the argument. When given a logarithmic equation \(\log_b a = c\), the base \(b\) is the number that is raised to the power \(c\) to get \(a\). The argument \(a\) is the result of this exponential expression. Recognizing these components is essential in evaluating or simplifying logarithms.
In the example \(\log_{27} 3 = \frac{1}{3}\), the base \(b\) is 27, and the argument \(a\) is 3. The number \(c\), which is \(\frac{1}{3}\), represents the power or exponent. Correctly identifying these parts of a logarithmic expression enables you to convert it into exponential form with confidence. By doing so, it becomes easier to solve or analyze the expression further if needed.

Problem-Solving Steps

Efficient problem-solving with logarithms requires a clear understanding of the process. Let’s go through the steps taken to convert the logarithmic equation \(\log_{27} 3 = \frac{1}{3}\) into its exponential form:

• **Step 1:** Identify the Components
  Recognize the base, argument, and result of the logarithm. In this case, the base \(b\) is 27, the argument \(a\) is 3, and the exponent or result \(c\) is \(\frac{1}{3}\).
• **Step 2:** Apply the Exponential Pattern
  Use the template \(b^c = a\) to set up the equation. Here, substitute the known values into the structure: \(27^{\frac{1}{3}} = 3\).

By understanding these steps, students can gain insight into the link between logarithms and exponentials. This approach not only clarifies the conversion process but also fortifies foundational math skills, vital for tackling more complex problems later on.