Problem 27 - Intermediate Algebra

Question

Write each logarithmic equation as an exponential equation. See Example 1. Do not solve. $$ \log _{4} \frac{1}{64}=-3 $$

Step-by-Step Solution

Answer

The equation \( \log_{4} \frac{1}{64} = -3 \) is equivalent to \( 4^{-3} = \frac{1}{64} \).

Step 1: Understand the Logarithmic Form

In the equation \( \log_b a = c \), \( b \) is the base of the logarithm, \( a \) is the argument, and \( c \) is the result of the logarithm. Here, the base \( b \) is 4, the argument \( a \) is \( \frac{1}{64} \), and the result \( c \) is -3.

Step 2: Convert to Exponential Form

To convert the logarithmic equation to an exponential form, use the relationship \( \log_b a = c \) which translates to \( b^c = a \). Here, you substitute \( b = 4 \), \( c = -3 \), and \( a = \frac{1}{64} \) to get the exponential form.

Step 3: Write the Exponential Equation

Using the information from Step 2, write the exponential equation \( 4^{-3} = \frac{1}{64} \). This demonstrates that 4 raised to the power of -3 equals \( \frac{1}{64} \).

Key Concepts

Logarithmic Form

Logarithms are a way of expressing numbers through exponents. The logarithmic form of an equation is essentially about finding what exponent you need to raise a base number to, in order to achieve a specific value. Let's break it down using the general form: If you have an equation \( \log_b a = c \), then \( b \) is known as the "base," \( a \) is the "argument," and \( c \) is the "result" or "exponent."

The equation \( \log_4 \frac{1}{64} = -3 \) follows this structure:

• Base: 4
• Argument: \( \frac{1}{64} \)
• Result: -3

This equation asks the question: "What power must 4 be raised to, in order to get \( \frac{1}{64} \)?" The answer is -3, which tells us everything we need about the structure and transformation of the logarithm into exponential form.

Exponential Form

Transforming a logarithmic equation into exponential form is straightforward once you understand the components. When you take a logarithm \( \log_b a = c \) and convert it to its exponential counterpart, you express that equation as \( b^c = a \). This conversion reveals the underlying power relation between these numbers.

For our example equation \( \log_4 \frac{1}{64} = -3 \), we translate it into its exponential form by rewriting the equation as \( 4^{-3} = \frac{1}{64} \). This tells us that when you raise 4 to the power of -3, you indeed get \( \frac{1}{64} \). Every log can be flipped into an exponent—this is the powerful essence behind logs and exponents being two sides of the same coin.

Base and Argument in Logarithms

Understanding the base and argument in a logarithm is key to mastering logarithmic expressions. Let's explore what these mean.

• Base: This is the number you are repeatedly multiplying in the exponential form. In our example \( \log_4 \frac{1}{64} = -3 \), the base is 4. In the exponential equivalent, 4 is the number that is being raised to the power specified by the result.
• Argument: The argument is what you achieve by raising the base to a specific exponent. Here, the argument is \( \frac{1}{64} \). In the exponential expression \( 4^{-3} = \frac{1}{64} \), \( \frac{1}{64} \) is reached by raising 4 to the power of -3.

A change in either the base or the argument would significantly alter the outcome of the logarithmic expression. It is this relationship between base, exponent, and argument that forms the core understanding necessary when transitioning from logarithmic to exponential forms.