Problem 55 - Algebra 2

Question

Evaluate each expression. $$ \log _{3} 81 $$

Step-by-Step Solution

Answer

The expression evaluates to 4.

Step 1: Understand the Logarithm

The expression \(\log _{3} 81 \) asks us to find which power the base 3 must be raised to, in order to result in 81. In other words, we need to solve for \( x \) in the equation \( 3^x = 81 \).

Step 2: Express 81 as a Power of 3

We need to express 81 as a power of 3. Starting from 3, we can multiply to see how many times we need to do so to reach 81: 3, 9, 27, 81. Thus, 81 can be expressed as \( 3^4 \).

Step 3: Solve the Logarithmic Equation

Since 81 is equal to \( 3^4 \), we can equate \( 3^x = 3^4 \). From the equation, we can see that \( x = 4 \). Therefore, \(\log _{3} 81 = 4 \).

Step 4: Verify the Solution

To verify, substitute \( x = 4 \) back into the equation \( 3^x = 81 \). That gives us \( 3^4 = 81 \), which is correct. Therefore, the solution \(\log _{3} 81 = 4 \) is verified.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable power or exponent. This changeable exponent is what provides exponential functions with their unique properties. For example, in the equation \(3^x\), the number 3 is the constant base, while \(x\) is the exponent that can vary. These functions are crucial as they model growth and decay processes in various fields, such as economics, biology, and physics.

One of the remarkable characteristics of exponential functions is their rapid growth. For instance, if you keep increasing the value of \(x\) in \(3^x\), you'll notice that the values increase very quickly. This rapid growth is due to the multiplicative nature of exponentiation. Each increment in \(x\) results in multiplying the previous result by the base. Understanding exponential functions provides a strong foundation for tackling logarithms, which are inversely related.

Base of a Logarithm

In the realm of logarithms, the base plays a crucial role. It is the number that is raised to a power to achieve a given result. In our specific example of \( \log _{3} 81 \), the base is 3. This tells us that we want to find out how many times we need to multiply 3 by itself to get 81. Essentially, you are determining the number of times the base must appear as a factor.

Every logarithm involves finding the power to which the base is raised. This is the opposite operation of exponentiation. It helps to think of a logarithm as an answer to a question: "To what power must the base be raised?" Knowing the base gives you half of the battle when dealing with logarithms, as it frames the entire calculation.

Power of a Number

The power of a number is a fundamental concept in mathematics. It represents the number of times a base number is multiplied by itself. For instance, when dealing with \(3^4\), you multiply 3 by itself four times (3 × 3 × 3 × 3). In this context, 4 is the exponent or power, and it tells us how many 3s are in the product.

Understanding powers is a key part of both exponential functions and logarithms. When you express a number, such as 81, in terms of powers, you are simplifying the logarithmic process. By identifying 81 as \(3^4\), you can more easily find the logarithm.

Powers are prevalent in everyday situations. In technology, they describe binary systems and data storage sizes. In physics, they describe growth rates and signal decays. Mastering this concept can therefore open up doors to understanding complex scientific and mathematical phenomena.