Problem 3 - Algebra A Combined Function

Question

Write each as an exponential equation. $$ \log _{3} \frac{1}{27}=-3 $$

Step-by-Step Solution

Answer

The exponential equation is \( 3^{-3} = \frac{1}{27} \).

Step 1: Understand the Logarithmic Form

The equation given is in logarithmic form: \( \log_{3} \frac{1}{27} = -3 \). In general, a logarithmic equation \( \log_b a = c \) is equivalent to an exponential equation \( b^c = a \).

Step 2: Identify the Base, Exponent, and Result

In the logarithmic equation \( \log_{3} \frac{1}{27} = -3 \), the base is 3, the result (which is inside the logarithm) is \( \frac{1}{27} \), and the exponent is -3.

Step 3: Write as an Exponential Equation

Using the equivalency \( \log_b a = c \Rightarrow b^c = a \), write the given logarithmic equation as an exponential equation. Substitute \( b = 3 \), \( c = -3 \), and \( a = \frac{1}{27} \) into the formula, to get the exponential form: \( 3^{-3} = \frac{1}{27} \).

Step 4: Verify the Exponential Form

Calculate \( 3^{-3} \) to verify the equation. We have:\[ 3^{-3} = \left(\frac{1}{3}\right)^3 = \frac{1}{3 \times 3 \times 3} = \frac{1}{27} \]This confirms that our exponential equation is correct.

Key Concepts

Logarithmic Equations

Logarithmic equations are expressions that relate a logarithm to a specific number. They are often in the form \( \log_b a = c \). Here, \( b \) represents the base, which is the number being repeatedly multiplied. The result \( a \) is the output of raising the base to the power of \( c \). In our exercise, we've looked at \( \log_{3} \frac{1}{27} = -3 \), which exemplifies how a logarithmic equation translates the exponential relationship between these numbers.
Logarithmic equations answer the question: "To what power must the base be raised to produce a specific number?" It's a way of undoing the exponential operation, revealing the exponent once we know the base and the result. Mastering logarithmic equations is crucial as they are widely used in science, engineering, and even in computing.

Converting Forms

Converting a logarithmic equation to its exponential form is a straight-forward process. Understanding this conversion is essential, as it often simplifies the equation and makes calculations easier.
To convert \( \log_b a = c \) into exponential form, we use the relationship: \( b^c = a \). This means that if you have a logarithmic equation, you can express it by raising the base \( b \) to the power of the result \( c \), achieving \( a \).
In our example, \( \log_{3} \frac{1}{27} = -3 \), the exponential form is \( 3^{-3} = \frac{1}{27} \). This conversion shows how multiplying 3 by itself negative three times results in \( \frac{1}{27} \), making the original logarithmic equation much clearer and easily verifiable.

Base and Exponent Identification

Before converting a logarithmic equation to its exponential form, understanding and identifying the base and exponent is critical. This is where many students need clarity at first.
In \( \log_b a = c \), the base \( b \) is the number that you raise to a power. The exponent \( c \) is the power to which the base must be raised to achieve \( a \), the number inside the logarithm.
In our exercise, \( \log_{3} \frac{1}{27} = -3 \), the structure is:

• Base \( b = 3 \)
• Exponent \( c = -3 \)
• Result \( a = \frac{1}{27} \)

Once you pinpoint these components, converting to exponential form becomes an intuitive process.

Verification Process

The verification process ensures the accuracy of the converted equation from logarithmic to exponential form. It involves calculating the result of the exponential equation to check if it aligns as expected.
In our example, for \( 3^{-3} = \frac{1}{27} \), verification requires calculating \( 3^{-3} \). By understanding negative exponents, which invert the base, you realize \( 3^{-3} \) is the same as \( \left( \frac{1}{3} \right)^3 \).
Calculating step-by-step:

• Compute \( \left( \frac{1}{3} \right)^3 \)
• This gives: \( \frac{1}{3 \times 3 \times 3} \)
• Thus, \( \frac{1}{27} \)

This confirms our exponential form \( 3^{-3} = \frac{1}{27} \) is correct, demonstrating a reliable and effective method for checking work.