Problem 66 - Algebra 2

Question

Solve each equation or inequality. Check your solutions. $$ \log _{4} r=3 $$

Step-by-Step Solution

Answer

The solution is \( r = 64 \).

Step 1: Understand the Equation

The given equation is \( \log_{4} r = 3 \). This equation is in logarithmic form and means the power to which the base 4 must be raised to obtain \( r \) is 3.

Step 2: Convert Logarithmic Form to Exponential Form

To solve \( \log_{4} r = 3 \), we convert the logarithmic equation to its equivalent exponential form. This is done by rewriting the equation as \( r = 4^3 \).

Step 3: Calculate the Exponential Form

We now calculate \( 4^3 \). This involves multiplying 4 by itself twice: \( 4 \times 4 \times 4 = 64 \). Thus, \( r = 64 \).

Step 4: Verify the Solution

To verify, substitute \( r = 64 \) back into the original equation: \( \log_{4}(64) = 3 \). Calculate by expressing 64 as a power of 4: \( 64 = 4^3 \), hence \( \log_{4}(64) = 3 \). The left-hand side equals the right-hand side, confirming our solution is correct.

Key Concepts

Exponential Form

Understanding the concept of exponential form is crucial when dealing with logarithmic equations, such as \( \log_{4} r = 3 \). The exponential form helps us uncover the values we are searching for in a straightforward manner.

To convert a logarithmic equation into an exponential form, we rearrange it. For our given equation, \( \log_{4} r = 3 \), it reads "the power to which 4 must be raised to result in \( r \) is 3". Transforming this into its exponential counterpart, \( r \) is expressed as \( r = 4^3 \).

• The base of the logarithm (4) becomes the base of the exponent.
• The result (\( r \)) is the same in both forms.
• The logarithmic value (3) becomes the exponent.

This conversion clarifies the exact number we are seeking without altering the equation’s integrity. It's like decoding a secret message from its tangled form into readable language.

Solution Verification

Solution verification ensures that the answer satisfies the original equation, proving that no mistakes were made along the way. For our log problem, we already calculated \( r = 64 \) from the exponential form \( 4^3 \).

Verification starts by substituting this back into the original logarithmic equation: \( \log_{4}(64) \). Now, we need to determine if this substitution indeed equals 3. Since it holds true that \( 64 = 4^3 \), our calculation matches the equation perfectly, verifying our solution.

• Substitute the found value back into the original equation.
• Convert the number back into its exponential form.
• Check if both sides equal, ensuring accuracy.

This step is vital; it confirms that our transformation and calculations haven't led us astray.

Base Conversion

In mathematical contexts, understanding how bases work is extremely helpful, particularly in logarithms and exponents. Base conversion often involves changing numbers into the same base, making it simpler to work through various calculations.

For the problem \( \log_{4} r = 3 \), no direct base conversion of the logarithm was needed since both sides naturally aligned. However, sometimes we might need to express a number using different bases to simplify calculations. Understanding base conversions can aid in these situations.

• Recognize the base of your logarithm or exponential term.
• Convert numbers to see if they can be expressed as a power of the same base.
• Ensure consistency in comparative contexts.

This knowledge can help streamline complex equations or compare different logarithmic expressions with different bases. Knowing how to manipulate bases helps maintain flexibility in problem-solving.