Problem 107 - Introductory and Intermediate Algebra for College Students 4th

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 logarithm of a product is equal to the sum of the logarithms of its factors.\( \log(MN) = \log(M) + \log(N) \)

Step 1: Graph of f(x) and g(x) for Subpart a

Using a graphing tool or calculator, plot the following two functions in the same view: \(f(x)=\ln(3x)\) and \(g(x)=\ln3 + \ln x\). Observe their intersection and behavior.

Step 2: Graph of f(x) and g(x) for Subpart b

Similarly, plot also \(f(x)=\log(5x^{2})\) and \(g(x)=\log5 + \log x^{2}\). Pay attention to any patterns or relationships between the two graphs.

Step 3: Graph of f(x) and g(x) for Subpart c

Finally, graph \(f(x)=\ln(2x^{3})\) and \(g(x)=\ln2 + \ln x^{3}\). Compare the results with the previous subparts.

Step 4: Observations

Reviewing the graphs plotted, it's clear that in each case \(f(x)=g(x)\). This means the functions are identical, leading to the generalization that \(\log(MN)= \log(M) + \log(N)\)

Step 5: Applying the Generalized Formula

To complete the final part of the exercise, use the derived generalized formula of the logarithm of a product. Thus, the logarithm of a product is equal to the sum of the logs of its factors.

Key Concepts

Properties of Logarithms

Logarithms have several useful properties that simplify complex mathematical operations. These properties are key for algebraic manipulation and problem-solving. One such property involves the sum, difference, and power of logarithms.

• Product Rule: The product rule states that the logarithm of a product is equal to the sum of the logarithms of its factors: \(\log_b(MN) = \log_b(M) + \log_b(N) \). This allows you to break down complex products into simpler components.


• Quotient Rule: Similarly, the logarithm of a quotient is the difference of the logarithms: \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\).


• Power Rule: In the power rule, if a number is raised to a power inside the log, you can move the exponent to the front: \(\log_b(M^p) = p \cdot \log_b(M)\).

These properties simplify calculations and allow easier manipulation of logarithmic expressions.

Graphing Logarithms

Graphing logarithmic functions involves plotting points on a coordinate plane. Since logarithms are the inverses of exponential functions, understanding their graph behavior can be very insightful. Logarithmic graphs are useful in visualizing how these functions behave.

• Shape and Direction: Logarithmic functions generally produce upward sloping curves that approach the vertical axis but never touch it.


• Domain: The domain of a logarithmic function is always positive real numbers, \((0, \infty)\), because you can't take the log of a non-positive number.


• Asymptotes: Logarithmic graphs have a vertical asymptote at \(x = 0\), meaning that as \(x\) approaches zero, the logarithmic value decreases to \(-\infty\).
• Intercepts: The typical logarithmic function \(\log_b(x)\) crosses the x-axis at \(x = 1\).

Practicing how to graph these functions with different bases can deepen your understanding of logarithmic behavior.

Logarithmic Equations

Solving logarithmic equations involves isolating the variable using the properties of logarithms. These types of equations are commonly encountered in algebra and calculus.First, you need to understand the basic form of a logarithmic equation: \(\log_b(x) = y\). To solve for \(x\), rewrite the equation in its exponential form: \(x = b^y\).

• Using Logarithmic Properties: Utilize properties like the product, quotient, and power rules to simplify the expressions.


• Isolating the Logarithm: Ensure that the logarithm is isolated on one side of the equation before attempting to solve it.


• Exponential Rewriting: Convert the logarithmic equation to an exponential one to solve for the variable easily.

Commonly, logarithmic equations might include other terms, and algebraic manipulation is necessary to isolate the logarithmic part first. Always check your solutions in the context of the initial equation to ensure no extraneous solutions are included.

Product Rule for Logarithms

The product rule for logarithms is a fundamental concept used to simplify complex logarithmic expressions. This rule comes in handy when dealing with products within a log function. The mathematical statement for the product rule is: \[\log_b(MN) = \log_b(M) + \log_b(N) \]

• Simplifying Expressions: The product rule allows for the decomposition of a single logarithm containing a product into the sum of two separate logs. This approach can simplify solving and manipulating such expressions.


• Verifying Equations: This rule is also used to verify some standard identities in log equations, like in the exercise, where \(f(x) = g(x)\) for functions \(f(x) = \ln(3x)\) and \(g(x) = \ln3 + \ln x\) show equality, proving \(\ln(3x) = \ln3 + \ln x\).


• Real-Life Applications: The product rule is practical not only in mathematics but also in fields like thermodynamics and acoustics where logarithms are frequently used to measure rates of change and intensity.

Understanding and being able to apply the product rule for logarithms can greatly simplify complex calculations and help solve logarithmic equations more efficiently.