Problem 133 - College Algebra

Question

In parts (a)-(c), graph \(f\) and \(g\) in the same viewing rectangle. a. \(f(x)=\ln (3 x), g(x)=\ln 3+\ln x\) b. \(f(x)=\log \left(5 x^{2}\right), g(x)=\log 5+\log x^{2}\) c. \(f(x)=\ln \left(2 x^{3}\right), g(x)=\ln 2+\ln x^{3}\) d. Describe what you observe in parts (a)-(c). Generalize this observation by writing an equivalent expression for \(\log _{b}(M N),\) where \(M>0\) and \(N>0\) e. Complete this statement: The logarithm of a product is equal to _____.

Step-by-Step Solution

Answer

The graphs of functions in parts (a), (b), and (c) coincide indicating that \(f(x) = g(x)\). As a result, \(\log_b(MN) = \log_bM + \log_bN\). Therefore, 'The logarithm of a product is equal to the sum of the logarithms'.

Step 1: Graphing functions in part (a)

Draw two logarithmic functions, first as \(f(x)= \ln (3x)\) and the second as \(g(x)= \ln 3 + \ln x \). You will observe that both graphs are the same, indicating \(f(x) = g(x)\).

Step 2: Graphing functions in part (b)

Similarly, draw two functions, \(f(x)= \log (5x^2)\) and \(g(x)= \log 5 + \log x^2\). We will see that both graphs align perfectly, meaning that \(f(x) = g(x)\).

Step 3: Graphing functions in part (c)

Continuing the process with the third set, we graph \(f(x)= \ln (2x^3)\) and \(g(x)= \ln 2 + \ln x^3\). Once more, we will see that the two functions coincide with each other. Thus, \(f(x) = g(x)\).

Step 4: Generalizing the observation

Realizing the pattern where each pair of f(x) and g(x) are equal, we can generalize that \(\log_b(MN) = \log_bM + \log_bN\), where M, N > 0. This is a fundamental property of logarithms indicating the logarithm of a product is the sum of the logarithms.

Step 5: Completing the statement

As we have established in the previous steps, we can say: The logarithm of a product is equal to the sum of the logarithms of the individual terms.

Key Concepts

Logarithmic Functions

Logarithmic functions are a crucial concept in mathematics, particularly in calculus and algebra. A logarithmic function is the inverse of an exponential function. This means that while an exponential function like \( b^x \) grows rapidly as \( x \) increases, a logarithmic function like \( \log_b(x) \) grows more slowly. Logarithms are useful in situations where we want to deal with very large numbers or quantities that change multiplicatively.

Key characteristics of logarithmic functions include:

• The domain of a logarithmic function is all positive real numbers because you can't take the log of zero or a negative number.
• The range is all real numbers because it can output any real value.
• Asymptotic behavior occurs as the input approaches zero from the positive side, meaning the graph approaches but never touches the y-axis.

Understanding these can be helpful when solving problems involving growth and decay, such as computing interest, population dynamics, or unraveling complex algebraic puzzles.

Product Rule of Logarithms

One of the key properties of logarithms is the product rule. This rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. Mathematically, this is expressed as:
\[\log_b(MN) = \log_bM + \log_bN\]This rule was demonstrated in the exercise where equivalent expressions were plotted on the same graph, revealing that:

• The log of a single product was equivalent to the sum of logs of each factor individually.
• This property holds true for natural logs and common logs alike, as seen in exercises with \( \ln (3x) \) and \( \log (5x^2) \).

In working with logarithms, applying this rule can simplify complicated expressions, making it easier to manipulate and solve equations. It also provides a shortcut in many mathematical calculations, allowing for the breakdown of a complex logarithmic expression into simpler parts.

Equivalent Expressions

When dealing with logarithms, expressions can be rewritten to take different forms while still maintaining equivalence. This is an important skill for simplifying mathematical problems, especially ones involving complex logarithmic terms. In the previous exercise, the concept of equivalent expressions was applied by recognizing that:

• \( f(x) = \ln (3 x) \) is the same as \( g(x) = \ln 3 + \ln x \).
• Similarly, \( f(x) = \log(5x^2) \) matches \( g(x) = \log 5 + \log x^2 \).

These pairs of equations are equivalent due to the product property of logarithms, showing that the operations inside the log can be split across sums outside it. Transforming expressions into equivalent forms can help reveal underlying patterns or simplify the approach to solving equations in calculus and algebra. Understanding equivalence in this context enhances computational flexibility and problem-solving efficiency.

Graphing Logarithmic Functions

Graphing logarithmic functions helps visualize how these equations behave across different conditions and inputs. In graphing the examples from the exercise, students observed how the same functions plotted in alternative forms yielded identical graphs. This reinforces the understanding of equivalence in practice.

Key elements when graphing logarithmic functions:

• The vertical asymptote at \( x = 0 \) as logs are undefined for non-positive numbers.
• The general shape is a curve increasing to the right, which never touches the y-axis.
• The growth is slower compared to linear or polynomial functions for the same range of input values.

In the exercise, functions like \( f(x) = \ln (3x) \) and \( g(x) = \ln 3 + \ln x \) visually confirmed that these were equivalent by perfectly overlaying on the graph. Graphing these functions allows students to see the theoretical foundations come to life in a tangible way, reinforcing abstract concepts through visual means. This practice is particularly helpful for students who benefit from visual learning styles.