Problem 28 - Intermediate Algebra

Question

Write each logarithmic equation as an exponential equation. See Example 1. Do not solve. $$ \log _{6} \frac{1}{36}=-2 $$

Step-by-Step Solution

Answer

The exponential form is \( 6^{-2} = \frac{1}{36} \).

Step 1: Introduce the Concept

Before we can convert a logarithmic equation to an exponential equation, we need to understand what each part of the logarithmic statement represents. The general form is \( \log_b a = c \), which means that \( b^c = a \). The base of the logarithm is \( b \), the argument is \( a \), and the result is \( c \).

Step 2: Identify Parts of the Equation

In the given equation \( \log_6 \frac{1}{36} = -2 \), identify the base \( b \), the argument \( a \), and the result \( c \). Here, \( b = 6 \), \( a = \frac{1}{36} \), and \( c = -2 \).

Step 3: Write the Exponential Form

Based on the identified parts of the equation, write the corresponding exponential form. Using the formula \( b^c = a \), substitute the values: \( 6^{-2} = \frac{1}{36} \). This converts the logarithmic equation into its exponential form.

Key Concepts

Exponential Equations

Exponential equations are equations where the variable is located in the exponent. They generally take the form \( b^c = a \), where \( b \) is the base, \( c \) is the exponent, and \( a \) is the result or outcome. These equations are fundamentally connected to real-world phenomena that exhibit rapid growth or decay, such as population growth and radioactive decay.
Understanding exponential equations is essential as they allow us to model and solve a wide variety of problems in science and engineering.

• Exponential equations show the power relationship between numbers.
• The exponent indicates how many times the base is multiplied by itself.
• They are often solved by finding common bases or using properties of exponents.

When working with exponential equations, it is useful to have a clear understanding of the rules that govern operations with exponents, such as the product of powers and power of a power rules.

Logarithmic to Exponential Conversion

The conversion between logarithmic and exponential forms is a key concept in mathematics. This conversion allows for easier manipulation and solution of equations based on what form is more convenient or intuitive. A logarithmic equation like \( \log_b a = c \) directly translates to an exponential equation \( b^c = a \).

• The base \( b \) of the logarithm becomes the base in the exponential equation.
• The result of the logarithmic equation \( c \) becomes the exponent in the exponential equation.
• The argument \( a \) in the logarithmic equation is the result of the exponential expression.

This conversion is crucial because logarithmic and exponential functions are inverse operations. Hence, converting between these forms can simplify complex mathematical expressions and aid in problem-solving.

Mathematical Notation

Mathematical notation is a system of symbols used to represent numbers, operations, and concepts. It provides a universal language for mathematicians around the world. Notation allows us to express and understand complex ideas more easily and precisely. In terms of logarithms and exponents:

• The symbol \( \log_b a \) represents a logarithm, indicating the power to which base \( b \) must be raised to yield \( a \).
• The exponential form \( b^c = a \) conveys that the base \( b \) raised to the power of \( c \) equals \( a \).
• Simple, clear notation helps in converting mathematical problems from one form to another efficiently.

Familiarity with this notation is essential for interpreting and solving equations and is an integral part of the learning process in mathematics. Recognizing these symbols and their relationships enables students to develop deeper mathematical understanding.