Problem 34 - Elementary and Intermediate Algebra

Question

Write each logarithmic equation as an exponential equation. See Example 1. Do not solve. $$ x=\log _{9} 81 $$

Step-by-Step Solution

Answer

The exponential form is \( 9^{x} = 81 \).

Step 1: Understanding the Logarithmic Equation

A logarithmic equation of the form \( x = \log_{b} a \) represents the power to which the base \( b \) must be raised to yield \( a \). In this problem, we have \( x = \log_{9} 81 \).

Step 2: Converting to the Exponential Form

To convert the logarithmic equation \( x = \log_{9} 81 \) to an exponential equation, identify the base \( 9 \), the exponent \( x \), and the result \( 81 \). The conversion yields the exponential equation \( 9^{x} = 81 \).

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical expressions that use logarithms to determine unknown values. A logarithm of a number is essentially the inverse operation of exponentiation. This means, if you have run across an equation like \( x = \log_{b} a \), it signifies that \( x \) is the exponent to which the base \( b \) must be raised to produce \( a \). Logarithmic equations help in solving for exponents when they are not easily discernible. To become comfortable with logarithms, remember:

• The base of the logarithm determines how you transform the equation.
• A logarithm answers the question: "To what power must the given base be raised, to achieve the given number?"

Think of a logarithm like this: If multiplying a number repeatedly by itself is like taking steps uphill, the logarithm is the reverse, like counting each step back down. Be sure to always identify the base and the result when working with logarithmic equations.

Exponential Equations

Exponential equations are equations where variables appear as exponents. An exponential equation converts a logarithmic equation into a more straightforward form, where you can clearly see the relationship between the base, exponent, and result. For example, in transforming \( x = \log_{9} 81 \) into an exponential form, you would express it as \( 9^{x} = 81 \).When handling exponential equations:

• The base of the equation remains consistent as shown in the logarithmic expression.
• The original logarithmic expression exponent becomes the exponent in the exponential form.
• The number on the right side of the logarithmic equation becomes what the base is raised to in the exponential equation.

Converting logs to exponential form can simplify complex calculations, making it easier to visualize the exponentiation process.

Base and Exponent Concepts

Base and exponent are fundamental concepts in both exponential and logarithmic forms. The base is the number that you keep multiplying, while the exponent tells you how many times you multiply the base by itself. In the context of \( x = \log_{9} 81 \):

• The base \( 9 \) indicates the number being repeatedly multiplied.
• The exponent \( x \) reveals how many times the base \( 9 \) must be used in multiplication to reach \( 81 \).

This relationship is key to understanding both exponential growth and decay processes. The fundamental understanding of base and exponentiation helps in comprehending the power relationship between numbers and interpreting real-world phenomena such as growth rates in populations, interest compounded over time, and even radiocarbon dating. Emphasize the meaning of the base and exponent to derive accurate calculations.