Problem 46 - Algebra 2 and Trigonometry

Question

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ \log _{3} 81 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

The problem requires us to evaluate the logarithm \( \log_{3} 81 \). This means we need to find what power of 3 gives us 81.

Step 2: Express as an Exponential Equation

We express the logarithmic equation \( \log_{3} 81 = x \) in its exponential form. This gives us the equation \( 3^x = 81 \). The task now is to find the value of \( x \).

Step 3: Express 81 as a Power of 3

We know that \( 81 \) can be expressed as a power of 3. By performing the calculations, we find that \( 3^1 = 3 \), \( 3^2 = 9 \), \( 3^3 = 27 \), and \( 3^4 = 81 \). Thus, \( 81 = 3^4 \).

Step 4: Solve for x

Since \( 81 = 3^4 \), from our equation \( 3^x = 3^4 \), it follows that \( x = 4 \). Thus, \( \log_{3} 81 = 4 \).

Step 5: Verify the Solution

To verify, we can substitute back into the original exponential form: \( 3^4 = 81 \). Since our calculation matches the original expression, our solution is verified.

Key Concepts

Exponential Equations

Exponential equations are expressions where a base number is raised to a variable exponent. These equations take the form \( b^x = n \), where \( b \) is the base and \( n \) is the number we are trying to equal. In the context of our exercise, understanding exponential equations becomes essential when converting logarithmic expressions to find solutions. For example, when dealing with \( \log_3 81 \), we set up an exponential equation as \( 3^x = 81 \). This tells us we're looking for some power \( x \) that when applied to 3, results in 81. Solving exponential equations often involves expressing both sides as powers of the same base to easily determine the exponent value.

Evaluate Logarithms

Evaluating logarithms is about determining the power needed to raise a base to reach a specific number. This is essentially what we do when we solve \( \log_3 81 \). To evaluate, we ask ourselves: "To what power must 3 be raised to get 81?" The solution involves recognizing patterns and might require rewriting numbers as powers of the base. In general, logarithms allow us to reverse the process of exponentiation, finding the exponent given the result and base. In practical terms, understanding how to evaluate logarithms helps solve problems involving growth, decay, and scaling processes in various scientific applications.

Powers of Numbers

Understanding the powers of numbers involves recognizing how many times a base number multiplies itself. This foundational concept is crucial for dealing with both logarithmic and exponential equations. For instance, knowing that \( 3^4 = 81 \) directly helps in solving \( \log_3 81 \). Powers of numbers can often be determined by sequential multiplication:

• \(3^1 = 3\)
• \(3^2 = 9\)
• \(3^3 = 27\)
• \(3^4 = 81\)

Having a good grasp of powers speeds up computations and helps to see solutions more clearly. This knowledge is especially useful in math and science, where quantifying objects often involves exponential or logarithmic transformations of numbers.