Problem 23 - Intermediate Algebra

Question

Write each logarithmic equation as an exponential equation. See Example 1. Do not solve. $$ \log _{3} 81=4 $$

Step-by-Step Solution

Answer

The exponential equation is \( 3^4 = 81 \).

Step 1: Identify the base of the logarithm

In the given logarithmic equation, \( \log_{3} 81 = 4 \), the base of the logarithm is \( 3 \). This is the number that is raised to a power to yield another number.

Step 2: Identify the logarithm result

In the equation \( \log_{3} 81 = 4 \), the result of the logarithm is \( 4 \). This means \( 3 \) raised to the power of \( 4 \) results in \( 81 \).

Step 3: Identify the number on the other side of the equation

The number inside the logarithm, known as the argument, is \( 81 \). This is the result when the base \( 3 \) is raised to the power \( 4 \).

Step 4: Rewrite the logarithmic equation as an exponential equation

Using the format \( \log_{b} x = y \) implies \( b^y = x \), rewrite the equation \( \log_{3} 81 = 4 \) in exponential form. Thus, it becomes \( 3^4 = 81 \).

Key Concepts

Logarithmic Equation

A logarithmic equation is a mathematical expression where a logarithm function sets the relationship between two numbers. It is particularly useful when you need to solve equations where the unknown is an exponent. In our example, the equation is \( \log_{3} 81 = 4 \). What this equation tells us is that the base number 3, when raised to the power that solves the logarithm, gives the result 81. In essence, a logarithm asks the question: "To what power must the base be raised to produce a certain number?"

• The format of a logarithmic equation is \( \log_{b} x = y \)
• "b" is the base of the logarithm, a key component
• "x" is the value you raise the base "b" to achieve
• The result "y" is the power that solves the logarithm

You use this setup to manipulate and understand numbers in powerful ways, aiding in sciences involving exponential growth or decay.

Base of Logarithm

The base of a logarithm forms the foundation of the relationship in a logarithmic equation. It indicates the number that is raised to a certain power. In our given equation \( \log_{3} 81 = 4 \), the base is 3. This means all calculations are centered around this number.

• The base determines the sense of scale of the equation
• A different base would completely change the outcome
• You can have natural logarithms with base \( e \approx 2.718 \)
• Or common logarithms where the base is 10

Choosing an appropriate base is crucial for easy calculation and interpretation of data. A good understanding of how the base functions will help simplify converting logarithmic equations into exponential form.

Exponential Form

Converting logarithmic equations to exponential form is a crucial step in solving many mathematical problems. This transformation reveals the direct relationship between the base and the exponent result. For the equation \( \log_{3} 81 = 4 \), rewriting it in exponential form yields \( 3^4 = 81 \).

• An exponential form is written as \( b^y = x \)
• Here, "b" is the base of the logarithm
• "y" is the power or exponent
• "x" is the number you obtain by raising the base to the power "y"

By converting a logarithmic equation to its exponential counterpart, you gain another perspective that can often make complex problems more accessible. It ties together the concepts of logarithms and exponents, demonstrating their interchangeability and complementarity in mathematics.