Problem 28 - Precalculus with Limits

Question

Using Properties of Logarithms In Exercises \(21-36\) , find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\log _{3}(-27)$$

Step-by-Step Solution

Answer

\( \log _{3}(-27) \) is undefined

Step 1: Analyze the problem

We are asked to find the exact value of \( \log _{3}(-27) \). We can see right away the input is -27, a negative number.

Step 2: Find the logarithm of the negative number

According to logarithm properties, there is no real number solution for the log of a negative number in the real number system. The reason for this is because the logarithm function with a positive base only produces results for positive numbers as their inputs.

Step 3: Statement of the Result

Since the input into this logarithmic function is -27, which is a negative number, \( \log _{3}(-27) \) is undefined in the real number system.

Key Concepts

Properties of Logarithms

Logarithms have several important properties that help us simplify expressions and solve equations. Here are some key properties you should know:

• Product Property: This property states that the logarithm of a product is the sum of the logarithms of the factors. Mathematically, \( \log_b (MN) = \log_b M + \log_b N \).
• Quotient Property: This states that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator, \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \).
• Power Property: This property indicates that the logarithm of a power is equal to the power times the logarithm of the base, \( \log_b (M^n) = n \cdot \log_b M \).

Understanding these properties makes it easier to manipulate and calculate logarithmic expressions.

Logarithmic Function

A logarithmic function is an important mathematical function connected to exponents. It answers the question: "To what power must the base be raised in order to yield a certain value?" For example, in the expression \( \log_3 (27) \), we ask: "What power must 3 be raised to, to get 27?" The answer is 3 because \( 3^3 = 27 \).

Logarithmic functions have specific characteristics:

• They are the inverse operations of exponential functions.
• They are only defined for positive real numbers in the context of a positive base.
• The graph of a logarithmic function is a curve that approaches the vertical axis but never touches it, known as a vertical asymptote.

Real Number System

The real number system includes all the numbers you encounter in everyday life, encompassing:

• Natural Numbers: Counting numbers like 1, 2, 3,...
• Whole Numbers: Natural numbers including zero.
• Integers: Positive and negative whole numbers, including zero.
• Rational Numbers: Numbers that can be expressed as the quotient of two integers, like fractions.
• Irrational Numbers: Numbers that cannot be precisely expressed as fractions, such as \( \pi \) and \( \sqrt{2} \).

The real numbers are vital, as they encompass all the numbers required for most calculations in algebra and calculus.

Negative Input

The concept of negative input in logarithmic functions is crucial because it deals with the limitations of logarithms. With a positive logarithmic base, the function accepts only positive real numbers as valid inputs. Logarithms of negative numbers are undefined in the real number system. This means:

• A logarithmic function like \( \log_3 (-27) \) is undefined because the logarithm of a negative number does not yield a real number result.
• The domain of a logarithmic function is restricted to positive numbers.
• This restriction aligns with how exponential functions grow, as they never produce negative results when the base is positive.

Understanding why logarithms of negative numbers are undefined helps to avoid errors in calculations and clarifies the limits of these mathematical operations.