Question

Graphical Analysis Use a graphing utility to graph \(f(x)=\ln 5 x\) and \(g(x)=\ln 5+\ln x\) in the same viewing window. What do you observe about the two graphs? What property of logarithms is being demonstrated graphically?

Short answer

The two graphs overlap and demonstrate the logarithmic product rule, \(\ln a+\ln b = \ln ab\), graphically indicating that the sum of two logarithms equals the logarithm of their product.

Step-by-step solution

Graph the First Function

Using the graphing tool, graph \(f(x) = \ln 5x\). To graph this, input the function into the graphing tool and analyze the graph.

Graph the Second Function

Next, graph \(g(x)= \ln 5+\ln x\). To do so, input this function into the graphing tool. When plotted on the same viewing window as the first function, notice any patterns and overlaps between the two graphs.

Compare and Identify the Logarithm Property

Observe and compare the two graphs. They will overlap precisely, demonstrating the property of logarithms that states \(\ln a+\ln b =\ln ab\). This property is known as the logarithmic product rule, which graphically shows that the sum of two logarithms equals the logarithm of their product.

Key concepts

Graphing Logarithmic Functions

Understanding how to graph logarithmic functions is crucial when learning about logarithms. A logarithmic function can be represented as \(f(x) = \log_b(x)\), where \(b\) is the base and \(x\) is the argument of the logarithm. To graph \(f(x)=\ln(5x)\), consider the natural logarithm with a base of \(e\). The curve passes through the point \((1, 0)\) because \(\ln(e^1) = 1\), and approaches negative infinity as \(x\) approaches zero from the right.

As you graph, observe the rate at which the function increases. Logarithmic functions grow slowly compared to polynomial or exponential functions. Ensure the graph is plotted correctly using a graphing utility, which provides a visual representation and helps solidify the understanding of logarithmic behavior.

Properties of Logarithms

Logarithms have special properties that make them easier to work with, especially when dealing with products, quotients, or powers. One fundamental property is the logarithmic product rule, which states that the log of a product is equal to the sum of the logs of its factors, shown as \(\ln(a) + \ln(b) = \ln(ab)\).

Another important property is the quotient rule: \(\ln(a) - \ln(b) = \ln(\frac{a}{b})\), and the power rule: \(\ln(a^x) = x\ln(a)\). These properties allow you to simplify and manipulate logarithmic expressions, which is essential in algebra and calculus. Understanding these properties also aids in the graphing process, as they can explain why certain logarithmic functions graph in the way they do.

Graphing Utility

A graphing utility is an invaluable tool for visualizing functions, particularly when dealing with logarithms. It allows students to enter any function and see its graph on a coordinate plane. To use a graphing utility effectively, input the correct function notation and adjust viewing windows to properly display the behavior of the function.

For example, graphing \(f(x) = \ln(5x)\) and \(g(x) = \ln(5) + \ln(x)\) side by side demonstrates that they are the same, highlighting the logarithmic product rule graphically. Exploring functions with a graphing utility not only helps in understanding the immediate problem but also reinforces general concepts in graphing and function analysis.

Comparing Functions

Comparing functions is a process of analyzing two or more functions to determine their similarities and differences. When you graph \(f(x)=\ln(5x)\) and \(g(x)=\ln(5)+\ln(x)\), you can visually compare these functions to verify their equivalence.

Both functions will produce the same graph, which is a clear indication of the logarithmic product rule at play. Observing the overlapping graphs reinforces the idea that the logarithm of a product is the sum of the logarithms. Comparing functions in this way helps students recognize patterns, predict behaviors, and ultimately gain a deeper understanding of how different functions, especially logarithms, relate to each other.