Problem 9 - Intermediate Algebra

Question

Fill in the blanks. \(\log _{6} 36=2\) means____ \(=\)_____

Step-by-Step Solution

Answer

6^2 = 36

Step 1: Understand Logarithm Definition

Logarithm is the inverse operation to exponentiation. The expression \( \log_b(a) = c \) signifies that \( b^c = a \).

Step 2: Identify Given Logarithmic Expression

We have \( \log_{6} 36 = 2 \). This tells us that when base 6 is raised to the power 2, the result is 36.

Step 3: Convert Logarithmic Equation to Exponential Form

From the definition, \( \log_{6} 36 = 2 \) can be rewritten as an exponential equation: \( 6^2 = 36 \).

Step 4: Fill in the Blanks

Use the exponential form derived to fill in the blanks: \( 6^2 = 36 \).

Key Concepts

Understanding Exponential Equations

In the realm of mathematics, exponential equations hold a significant place. They involve expressions where a constant base is raised to a certain power or exponent. An exponential equation is generally represented as \(b^c = a\), where \(b\) is the base, \(c\) is the exponent, and \(a\) is the resulting value.
These equations might seem complex initially, but they are utilized in various real-world applications including calculating compound interest and modeling population growth. In our example, we converted the logarithmic statement \(\log_{6} 36 = 2\) to an exponential form \(6^2 = 36\), which shows the exponential nature of logarithms.

• The base \(b\) in \(6^2\) is 6.
• The exponent \(c\) is 2, indicating how many times the base is multiplied by itself.
• The result, which is 36, is the number obtained after performing the exponentiation.

Understanding exponential equations gives us the power to decode other mathematical ideas such as logarithms.

Inverse Operations Explained

Inverse operations are mathematical operations that undo each other. They are like mathematical opposites. For instance, addition and subtraction are inverse operations. This means if you add a number and then subtract the same number, you’re back where you started.
When it comes to logarithms and exponentials, these also form a pair of inverse operations:

• Exponentiation involves raising a base to a power.
• Logarithms help us find that power when the base and resultant number are known.

In simple terms, if logarithms help us find \(c\) in \(\log_b(a) = c\), exponential operations state \(b^c = a\).
These inverse operations are particularly useful for solving equations and help us switch between exponential and logarithmic forms, as seen in the exercise when transitioning from \(\log_{6} 36 = 2\) to \(6^2 = 36\). This ability to switch aids in various calculations, transforming complex problems into simpler computations.

Base and Exponent Fundamentals

In any power expression, understanding the base and exponent is crucial. The base is the number being multiplied, and the exponent indicates how many times the base is used as a factor.
For example, in the expression \(6^2\), 6 is the base, and 2 is the exponent, directing us to multiply 6 by itself once to obtain 36.
Some essential facts about base and exponent:

• The base must be a positive number for typical calculations. In more advanced topics, negative bases can also appear.
• The exponent dictates the number of times the base is used in multiplication. When the exponent is 2, we square the base.

Resolving such expressions can deepen one's comprehension of logarithms' role as inverse operations. In logarithmic terms, the expression \(\log_{6} 36 = 2\) is understood by recognizing that the base (6) raised to the power of the exponent (2) results in 36.
Understanding these elements equips students to tackle more advanced math problems with confidence.