Problem 3 - Calculus and its applications

Question

Write an equivalent exponential equation. $$ \log _{27} 3=\frac{1}{3} $$

Step-by-Step Solution

Answer

The equivalent exponential equation is \( 27^{\frac{1}{3}} = 3 \).

Step 1: Understanding the Logarithmic Equation

The logarithmic equation given is \( \log_{27} 3 = \frac{1}{3} \). This indicates that 27 raised to the power of \( \frac{1}{3} \) equals 3.

Step 2: Rewrite as Exponential Form

Based on logarithmic properties that say \( \log_b a = c \) is equivalent to \( b^c = a \), rewrite the equation as an exponential form. Hence, change \( \log_{27} 3 = \frac{1}{3} \) to \( 27^{\frac{1}{3}} = 3 \).

Key Concepts

Logarithmic Functions

Logarithmic functions are powerful mathematical tools used to solve equations involving exponents. Essentially, a logarithm answers the question, "To what power must we raise a certain base to obtain a given number?" For instance, if we have the expression \( \log_{27} 3 \), it asks us to find the power we need to raise 27 to, in order to get 3. Here, the solution is \( \frac{1}{3} \), indicating that by raising 27 to the power of \( \frac{1}{3} \), we achieve 3. This function is the inverse of an exponential function and provides a way to deal with multiplicative problems that are difficult to solve directly. Logarithms can simplify complex equations and are often used in various fields, including science, engineering, and finance. They come in different bases, with base 10 (common logarithm) and base \( e \) (natural logarithm) being the most frequent.

Exponential Functions

Exponential functions are the flip side of logarithmic functions. While logarithms focus on finding the exponent, exponential functions deal with the expression directly. An exponential function is characterized by its constant base raised to a variable exponent, written as \( b^x \). In our exercise, when we convert \( \log_{27} 3 = \frac{1}{3} \) into its exponential form, we get \( 27^{\frac{1}{3}} = 3 \). This illustrates how these functions work.

• The base in this equation is 27,
• \( \frac{1}{3} \) is the power or exponent,
• and the result is 3.

Exponential functions are incredibly important in modeling growth and decay processes, such as population growth, radioactive decay, and even in calculating compound interest. Since they involve rapid increases or decreases, understanding exponentials is crucial in both theoretical and applied mathematics.

Basic Logarithmic Properties

Understanding the basic properties of logarithms can immensely simplify working with these expressions and equations. One fundamental property is based on the relationship between logarithms and exponentials. For any positive number \( b \), if \( \log_b a = c \), then the exponential form is \( b^c = a \). This was directly applied in the exercise, where \( \log_{27} 3 = \frac{1}{3} \) converts to \( 27^{\frac{1}{3}} = 3 \).

• 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: \( \log_b (x^y) = y \cdot \log_b x \)

These properties help to break down complex logarithmic expressions into simpler terms, easing computation. They are valuable when solving complex problems, making the manipulation of logarithmic equations less daunting and more systematic. Mastering these properties enhances overall fluency in mathematics, giving students the tools needed to tackle advanced concepts with ease.