Math Example: Graphs of Logarithmic Functions: Example 14

Topic

Logarithmic Functions

Description

This example presents the graph of the logarithmic function y = log₁₀(4x - 5) - 5. The graph opens upward, demonstrating how the combination of multiplication and subtraction inside the logarithm, along with subtraction outside the logarithm, affect the function's behavior. The domain of this function is x > 1.25, as the expression inside the logarithm must be positive.

Logarithmic functions are essential in various fields, including science, engineering, and finance. They are particularly useful for modeling phenomena that exhibit exponential growth or decay. This collection of examples helps teach logarithmic functions by presenting a variety of equations with different transformations, allowing students to visualize and understand how these changes affect the graph's shape, position, and behavior.

Providing multiple worked-out examples is crucial for students to fully grasp the concept of logarithmic functions. Each example builds upon previous ones, introducing new transformations and combinations of 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.

Teacher's Script: Now, let's examine our fourteenth example, y = log₁₀(4x - 5) - 5. Compare this graph to the previous example. Both have subtraction outside the logarithm, but this one also has subtraction inside. How does this change the domain of the function? Notice that the domain is now x > 1.25. Can you explain why? The subtraction of 5 outside the logarithm shifts the graph vertically downward. How does the combination of 4x - 5 inside the logarithm affect the graph's horizontal position? Can you identify where this graph intersects the y-axis? Is there a y-intercept in this case? If not, why?

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