Problem 102 - College Algebra

Question

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). \(\begin{array}{llllll}\text { Generalize } & \text { this } & \text { observation } & \text { by } & \text { writing } & \text { an }\end{array}\) 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.

Step 1: Graphing the Functions: Part a

To start the exercise, graph the functions \(f(x) = \ln(3x)\) and \(g(x) = \ln3 + \lnx\). It's important to remember that the base logarithm of a product is the sum of the logarithms of the factors.

Step 2: Graphing the Functions: Part b

Next, graph the functions \(f(x) = \log(5x^2)\) and \(g(x) = \log5 + \log x^2\). The properties of logarithms still apply here, even when dealing with a different base and including an exponent in the function.

Step 3: Graphing the Functions: Part c

Finally, graph the functions \(f(x) = \ln(2x^3)\) and \(g(x) = \ln2 + \lnx^3\). This last set of functions includes another property of logarithms, which allows us to bring the exponent out front in the logarithm.

Step 4: Describing Observations

For each of the graphed pairs above, observe that all the graphs are identical. They both represent the same function - a fundamental property of logarithms.

Step 5: Generalizing Observation

The observation from the previous steps leads to a general rule: the logarithm of a product is equal to the sum of the logarithms of the numbers being multiplied. This can be formally written as \(\log_b(MN) = \log_bM + \log_bN\), where \(M>0\) and \(N>0\)

Step 6: Completing the statement

Based on the observation and generalization, complete the statement as follows: The logarithm of a product is equal to the sum of the logarithms.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and are defined by the equation: \( y = \text{log}_b(x) \) where \( b \) is the base, \( x \) is the argument, and \( y \) is the exponent to which the base must be raised to obtain the value of \( x \). For any base \( b > 0 \) (and \( b eq 1 \) ), there's a corresponding logarithmic function. The natural logarithm function, typically written as \( y = \text{ln}(x) \) or \( y= \text{log}_e(x) \), uses Euler's number \( e \) as the base and is particularly important in calculus due to its unique properties.

Graphically, logarithmic functions have distinct characteristics. They pass through the point (1, 0), approach the y-axis as a vertical asymptote, and are undefined for \( x \leq 0 \). Because they are inverses, the graph of a logarithmic function is a reflection of its corresponding exponential function across the line \( y=x \). Logarithmic scaling can help with data that cover a wide range of values since it can compress large scales and expand small scales, making patterns more visible.

Graphing Logarithms

Graphing logarithmic functions requires understanding their behavior and key properties. As observed in the textbook exercise, logarithms of products, \( f(x) = \text{log}(MN) \) can be separated into the sum of the logarithms of the individual factors, \( g(x) = \text{log}(M) + \text{log}(N) \). This characteristic means that when graphing \( f(x) \) and \( g(x) \) from the exercise's parts (a)-(c), we end up with the same curve for both equations for any positive \( M \) and \( N \).

When graphing, note also that logarithmic functions increase slowly for values of \( x \) greater than 1 and decrease rapidly for values of \( x \) between 0 and 1. They never touch the y-axis or the negative side of the x-axis, as logarithms are not defined for non-positive values of \( x \). Learning to graph logarithms typically involves understanding their asymptotes, intercepts, and the effects of transformations such as vertical and horizontal shifts.

Logarithm Laws

The logarithm laws are vital for simplifying and manipulating expressions involving logarithms. A basic understanding of these laws can greatly help to solve logarithmic equations and to graph logarithmic functions. These laws include the product rule, quotient rule, and power rule, among others.

• Product Rule: \( \text{log}_b(MN) = \text{log}_b(M) + \text{log}_b(N) \) - This rule states that the logarithm of a product is the sum of the logarithms of the factors, as demonstrated in the exercise and solution provided.
• Quotient Rule: \( \text{log}_b(\frac{M}{N}) = \text{log}_b(M) - \text{log}_b(N) \) - The logarithm of a quotient is equal to the difference of the logarithm of the numerator and the logarithm of the denominator.
• Power Rule: \( \text{log}_b(M^p) = p \text{log}_b(M) \) - This law allows you to move the exponent of the argument to the front of the logarithm, making it easier to solve equations with logarithms of exponential expressions.

Understanding and applying these laws are crucial in algebra and higher-level math courses, as they simplify and expedite the process of solving problems involving logarithms.