Math Example: Graphs of Logarithmic Functions: Example 22

Topic

Logarithmic Functions

Description

This example presents the graph of the natural logarithmic function y = ln(3x) - 5. The graph opens upward, curving from left to right, demonstrating how the combination of multiplication inside the logarithm and subtraction outside affects the function's behavior. The domain of this function remains x > 0, as the expression inside the natural logarithm must be positive.

Natural logarithmic functions with various transformations are essential in many scientific and mathematical applications, particularly in calculus and differential equations. They are often used to model complex growth and decay processes in fields such as biology, physics, and economics. This collection of examples helps teach logarithmic functions by presenting a range of equations with different parameters, allowing students to visualize and understand how these changes impact the graph's shape, position, and range.

Providing multiple worked-out examples is crucial for students to fully grasp the concept of logarithmic functions and their transformations. Each example builds upon previous ones, introducing new variations and operations. This approach helps students recognize patterns, understand the effects of different parameters, and develop the ability to predict how changes in the equation will impact the graph of logarithmic functions.

Teacher's Script: Now, let's examine our twenty-second example, y = ln(3x) - 5. Compare this graph to the previous natural logarithm function we just saw. What are the main differences you notice? The domain remains x > 0, but how has the range changed? Consider how the multiplication by 3 inside the logarithm affects the graph's horizontal position. Then, think about how the subtraction of 5 outside the logarithm impacts the vertical position. Can you predict where this graph will intersect the y-axis? How does this transformation impact the overall behavior of the function compared to the previous example with addition outside the logarithm?

For a complete collection of math examples related to Logarithmic Functions click on this link: Math Examples: Graphs of Logarithmic Functions Collection.