Problem 23 - Intermediate Algebra

Question

For Problems \(21-40\), evaluate each expression. $$ \log _{3} 81 $$

Step-by-Step Solution

Answer

\(\log_{3} 81 = 4\)

Step 1: Understand the Problem

We need to evaluate the logarithm \( \log_{3} 81 \). This means we want to find the exponent \( x \) such that \( 3^x = 81 \).

Step 2: Express 81 as a Power of 3

To solve for \( x \), express 81 as a power of 3. We know that \( 81 = 3^4 \) since \( 3 \times 3 \times 3 \times 3 = 81 \).

Step 3: Set Up the Equation

Substitute the expression from the previous step into the logarithmic equation. This gives us \( \log_{3} 81 = \log_{3} (3^4) \).

Step 4: Apply Logarithmic Identity

Use the identity \( \log_b (b^a) = a \). Applying this identity, we get \( \log_{3} (3^4) = 4 \).

Step 5: Conclusion

We have found that the value of \( \log_{3} 81 \) is 4, which means 3 raised to the power of 4 equals 81.

Key Concepts

Evaluating Logarithms

When evaluating logarithms, we are essentially trying to find the exponent needed to raise a base to get a certain number. In the provided exercise, we are tasked with evaluating \( \log_{3} 81 \). This process involves understanding what exponent of the base (in this case, 3) results in the number 81. The logarithmic expression \( \log_{3} 81 \) can be translated into the equation \( 3^x = 81 \) where \( x \) is the unknown exponent. By solving this equation, we discover the value of \( x \) that makes the equation true.

Logarithmic Identities

Logarithmic identities are crucial for simplifying and solving logarithmic expressions. One key identity applied in the original solution is \( \log_b (b^a) = a \). This identity tells us that if our logarithm's base and the base of the exponential number inside the log match, we can directly take out the exponent. In our example, once we express 81 as \( 3^4 \), the identity allows us to simplify \( \log_{3} (3^4) \) directly to 4. This step is indispensable when working with logarithms as it can significantly simplify complex logarithmic calculations.

Properties of Exponents

The relationship between logarithms and exponents is deeply connected through their properties. Understanding the basic properties of exponents is essential when working with logarithmic expressions. In this task, we transformed 81 into \( 3^4 \) because it simplifies the logarithmic evaluation. The rule applied here indicates that multiplying the base number (3) by itself four times results in 81. Knowing how to express a number like 81 as an exponential form \( 3^4 \) is key to solving \( \log_{3} 81 \). This involves identifying powers of numbers, which helps to streamline both logarithmic and algebraic solutions.