Problem 25 - Algebra 2

Question

Write each equation in exponential form. \(\log _{4} \frac{1}{4}=-1\)

Step-by-Step Solution

Answer

The equation in exponential form is \( \frac{1}{4} = 4^{-1} \).

Step 1: Understand Logarithmic Form

The logarithmic form of an equation is written as \( \log_b a = c \), where \( b \) is the base, \( a \) is the result, and \( c \) is the exponent. In this exercise, \( \log _{4} \frac{1}{4}=-1 \) is provided in logarithmic form.

Step 2: Identify the Components

In the equation \( \log _{4} \frac{1}{4}=-1 \), the base \( b \) is 4, the result \( a \) is \( \frac{1}{4} \), and the exponent \( c \) is -1.

Step 3: Write the Exponential Form

The exponential form is determined from the logarithmic form by expressing it as \( a = b^c \). Thus, from \( \log _{4} \frac{1}{4}=-1 \), we write \( \frac{1}{4} = 4^{-1} \).

Key Concepts

Logarithmic Form

Logarithmic form is a way of expressing a number that helps us understand exponential relationships. In general, when you see an expression like \( \log_b a = c \), it is in logarithmic form. Here, \( b \) is called the base, \( a \) is the result, and \( c \) is the exponent. This form asks the question: "To what power must the base \( b \) be raised to produce \( a \)?"

Logarithms simplify expressions involving powers and are used in many fields such as mathematics, science, and engineering. They allow us to solve exponential equations and make certain types of data easier to work with. In the exercise, \( \log _{4} \frac{1}{4} = -1 \) tells us that \( 4 \) raised to what power equals \( \frac{1}{4} \). The answer, which is \(-1\), is the exponent in this case.

Base and Exponent

Understanding the terms base and exponent is crucial when dealing with logarithms and exponents. The base is the number that is repeatedly multiplied. In the case of exponential expressions and forms, it's the starting point from which powers are calculated.

For the equation \( \log _{4} \frac{1}{4} = -1 \):

• The base \( b \) is 4. It's the number we're repeatedly multiplying.
• The exponent \( c \) is -1, representing how many times the base will be used in a multiplication or division process.

In exponential form, these components play specific roles. The exponent tells us how many times to multiply the base by itself. If it’s negative, as in our example, it indicates division or a reciprocal action. Thus, \( 4^{-1} \) indicates that 4 should be taken as the reciprocal, which results in \( \frac{1}{4} \). This clarifies how exponents can also represent negative powers and not just positive multiplication.

Converting Logarithms to Exponents

Converting logarithmic equations to exponential form helps us visualize and solve them more effectively. In our given example \( \log _{4} \frac{1}{4} = -1 \), we want to express this in exponential terms.

The logarithmic form \( \log_b a = c \) can be transformed into exponential form with \( a = b^c \). To convert:

• Identify \( b \), \( a \), and \( c \) from the logarithm which are the base, result, and exponent respectively.
• Rearrange the equation to correspond with the format \( b^c = a \).

For our specific equation, \( \log _{4} \frac{1}{4} = -1 \), converting it gives us \( \frac{1}{4} = 4^{-1} \). This shows the direct relationship between the numbers in both forms: 4 raised to negative one equals \( \frac{1}{4} \). Practicing this conversion backward and forward helps strengthen our understanding of the relationship between logarithms and exponents.