Problem 9 - College Algebra and Calculus: An Applied Approach

Question

Use the definition of a logarithm to write the equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3$$81^{1 / 4}=3\)

Step-by-Step Solution

Answer

The logarithmic form of the equation \(81^{1/4}=3\) is \(\log _{81} 3=1/4\).

Step 1: Understanding the Exponential Form

First, you need to recognize the different parts of the exponential equation, \(81^{1/4}=3\). Here, the base is 81, the exponent is 1/4, and the result of this operation is 3.

Step 2: Rewriting in Logarithmic Form

Now, you apply the transformation of the exponential equation to its equivalent logarithmic equation. According to logarithm rules, the base of the exponential form becomes the base of the logarithm, the result of the exponential form becomes the argument of the logarithm, and the exponent of the exponential form becomes the result of the logarithm. So, our equation transforms to \(\log _{81} 3=1/4\).

Key Concepts

Exponential Equation

An exponential equation involves terms where variables appear as exponents. In the equation \(81^{1/4}=3\), 81 is the base, \(1/4\) is the exponent, and 3 is the result of raising the base to the given exponent. These types of equations exhibit exponential growth or decay, depending on the values involved.
Learning to recognize and understand the components of exponential equations is essential. It helps in transforming them into other forms, like logarithmic form. When you have an equation like \(a^{b}=c\), this tells you that when the base \(a\) is raised to the power \(b\), you get \(c\) as the result. Let's break down this concept further:

• Base: The number that is raised to a power (81 in our example).
• Exponent: The power to which the base is raised (\(1/4\) here).
• Result: The outcome of the base raised to the exponent (3 in our example).

Understanding these components is crucial for converting exponential expressions into logarithmic form, which is often necessary when solving for unknowns in equations or transforming between exponential and logarithmic equations.

Logarithm Rules

Logarithm rules are guidelines used to manipulate logarithmic expressions. They are especially useful when converting an exponential equation into logarithmic form. With these rules, you can understand the relationship between exponential and logarithmic forms.
The main rules you need to remember are:

• Product Rule: \(\log_b(mn) = \log_b(m) + \log_b(n)\).
• Quotient Rule: \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\).
• Power Rule: \(\log_b(m^n) = n\log_b(m)\).
• Conversion Rule: An exponential equation \(a^b = c\) can be written in logarithmic form as \(\log_a(c) = b\).

Applying these rules helps convert between the forms correctly. In our example, \(81^{1/4} = 3\) becomes \(\log_{81}(3) = 1/4\) using the conversion rule. Recognizing and using these rules correctly ensures accuracy when solving or simplifying logarithmic equations.

Base of the Logarithm

The base of a logarithm is a fundamental part of the logarithmic expression. In the example \(\log_{81}(3) = 1/4\), 81 is the base of the logarithm. It signifies that we are dealing with powers of 81.
The base of a logarithm determines the ‘scale’ in which the logarithm operates. It answers the question, "To what power must the base 81 be raised to yield 3?" In logarithmic form:

• Exponentiation Basis: The number that is consistently multiplied to reach another number.
• Common Base: A common logarithm has base 10, written as \(\log(\ldots)\).
• Natural Base: A natural logarithm has base \(e\), written as \(\ln(\ldots)\).

This understanding is crucial when interpreting and converting between exponential and logarithmic forms. The base dictates how you approach calculations and transformations, reinforcing the relationship between the exponential and logarithmic expressions.