Problem 14 - A Graphical Approach to College Algebra

Question

For each statement, write an equivalent statement in exponential form. $$\log _{4} \frac{1}{64}=-3$$

Step-by-Step Solution

Answer

The equivalent exponential form is \( 4^{-3} = \frac{1}{64} \).

Step 1: Understanding the Logarithmic Expression

The expression given is \( \log_{4} \frac{1}{64} = -3 \). This means that the logarithm of \( \frac{1}{64} \) with base 4 equals -3. Our task is to convert this logarithmic expression into an equivalent exponential form.

Step 2: Base and Result Identification

In any logarithmic expression \( \log_{b} a = c \), the base \( b \) raised to the power of \( c \) results in \( a \). Here, our base is 4, and \( c \) is -3, with \( \frac{1}{64} \) being the result.

Step 3: Applying the Exponential Rule

Using the rule from the previous step, we transform the expression \( \log_{4} \frac{1}{64} = -3 \) into exponential form by writing it as \( 4^{-3} = \frac{1}{64} \). This matches our base-exponent-logarithm relationship.

Key Concepts

Logarithmic Expression

A logarithmic expression is a powerful way to explore relationships between numbers. In general, a logarithmic expression is written as \( \log_{b} a = c \), where \( b \) is known as the base and \( a \) is the number we are taking the logarithm of, which gives us the result \( c \). This expression asks the question: "To what power should the base be raised to obtain the number \( a \)?"
For example, in the expression \( \log_{4} \frac{1}{64} = -3 \), the base is 4. We are finding the power that 4 must be raised to in order to yield \( \frac{1}{64} \). The answer is \(-3\), indicating that 4 raised to the power of \(-3\) results in \( \frac{1}{64} \).
Logarithmic expressions can be tricky at first, but they become more straightforward once you understand how the numbers relate. They allow us to compactly express relationships involving exponential growth and decay.

Base and Result Identification

Identifying the base and the result in a logarithmic expression is crucial for converting it into exponential form. The base is the constant number that is raised to a power. In the expression \( \log_{4} \frac{1}{64} = -3 \), the base is 4. This number tells us which number is being exponentiated.
The result is the number we want to achieve by raising the base to the given power. Here, \( \frac{1}{64} \) is the result of the expression. It is what we obtain when the base 4 is raised to the power of \(-3\). Therefore, in any logarithmic expression \( \log_{b} a = c \), it is essential to identify:

• \( b \) as the base
• \( a \) as the result
• \( c \) as the exponent that results in \( a \)

This understanding is key to translating a logarithmic statement into its corresponding exponential form.

Exponential Rule

The exponential rule allows us to translate a logarithmic expression into an exponential one. This rule states that for any logarithmic expression \( \log_{b} a = c \), it can be rewritten in exponential form as \( b^{c} = a \).
In practice, the exponential rule helps in see the same mathematical relationship from a different perspective. Using our example \( \log_{4} \frac{1}{64} = -3 \), we apply this rule and express it in exponential form as \( 4^{-3} = \frac{1}{64} \).
Steps to follow:

• Recognize the base (\( b \)) in the logarithm, which is 4 in this case.
• Identify the exponent (\( c \)), which is \(-3\).
• Apply the exponential rule to write it as \( b^{c} = a \) or \( 4^{-3} = \frac{1}{64} \).

The exponential rule is a handy guideline to convert between logarithmic and exponential forms, enabling us to solve and understand a wide range of algebraic problems.