Problem 39 - 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 Expression

The problem at hand is to evaluate the logarithmic expression \( \log_{3} 81 \). This means we need to find the exponent to which the base 3 must be raised to yield 81.

Step 2: Express 81 as a Power of 3

81 can be expressed as a power of 3. We know that 3 multiplied by itself four times equals 81: \[ 3^4 = 81 \].

Step 3: Apply the Definition of Logarithms

By the definition of logarithms, \( \log_{b} a = c \) means that \( b^c = a \). Therefore, for \( \log_{3} 81 \), we have \( 3^c = 81 \). Since \( 3^4 = 81 \), it follows that \( c = 4 \).

Step 4: Conclude the Evaluation

Based on the calculations and using the definition of logarithms, we determine that \( \log_{3} 81 = 4 \).

Key Concepts

Evaluating Logarithms

Evaluating logarithms is about finding the power to which a base number must be raised to produce a given number. In our example, the expression \( \log_{3} 81 \) is asking: "To what power should 3 be raised to get 81?". Think of a logarithm as the inverse of exponentiation. Just like multiplication is the opposite of division, logarithms undo what exponentiation does. It's like asking, "How many times do I need to use this base in a multiplication to get the number I want?"

• Identify the base: In our case, it is 3.
• Identify the number you're seeking to convert: Here, it is 81.
• Find the exponent: Determine how many times the base must be multiplied by itself to reach the desired number. For this, 3 raised to the power of 4 equals 81, so \( \log_{3} 81 = 4 \).

Breaking down logarithms into these steps helps in understanding their evaluation. It's all about recognizing the repeated multiplication and applying the definition of a logarithm.

Logarithmic Functions

Logarithmic functions are a type of function that uses logarithms. These functions are crucial in many scientific and real-world applications, such as measuring the pH in chemistry, the Richter scale for earthquakes, or the decibel scale for sound intensity. Logarithmic functions, generally written as \( f(x) = \log_{b}(x) \), involve a base \( b \) and an argument \( x \). When dealing with these functions, it's important to understand their graphical representation and behavior.Characteristics of logarithmic functions include:

• They have a vertical asymptote at \( x = 0 \), meaning the graph approaches this line but never touches it.
• These functions are defined only for positive values of \( x \).
• Their graphs pass through the point \( (1,0) \), because \( \log_{b}(1) = 0 \) regardless of the base.
• A logarithmic curve is generally increasing if \( b > 1 \), showing a gradual rise.

Understanding these characteristics helps predict the behavior and application of logarithmic functions in various practical scenarios.

Properties of Exponents

Exponents are fundamental in evaluating logarithms because they're essentially about multiplication done repeatedly. The properties of exponents allow you to rewrite and simplify expressions. Let's explore some core properties used in evaluating logarithmic expressions:

• Product of Powers: \( a^m \times a^n = a^{m+n} \). This property shows how multiplying powers of the same base adds their exponents.
• Power of a Power: \( (a^m)^n = a^{m \times n} \). When raising an exponent to another power, multiply the exponents.
• Power of a Product: \( (ab)^n = a^n \times b^n \). This allows distributing the exponent across all elements within the bracket.

These properties are extremely helpful during the process of expressing numbers like 81 as powers of 3, as we did when \( 3^4 = 81 \). Recognizing how these exponent rules apply makes the finding of logarithmic solutions much easier.