Math Example: Graphs of Logarithmic Functions: Example 11

Topic

Logarithmic Functions

Description

This example illustrates the graph of the logarithmic function y = log₁₀(3x + 9) + 1. The graph opens upward, demonstrating how the combination of multiplication and addition inside the logarithm, along with addition outside the logarithm, affect the function's behavior. The domain of this function is x > -3, as the expression inside the logarithm must be positive.

Logarithmic functions play a crucial role in various mathematical and scientific applications, from modeling exponential growth and decay to solving complex equations. This collection of examples helps teach logarithmic functions by presenting a range of equations with different parameters and transformations, allowing students to visualize and understand how these changes impact the graph's shape, position, and properties.

Exploring multiple worked-out examples is vital for students to develop a comprehensive understanding of logarithmic functions. Each example builds upon previous knowledge, introducing new concepts and transformations. This approach helps students recognize patterns, make connections between different forms of logarithmic equations, and develop problem-solving skills that can be applied to more complex scenarios.

Teacher's Script: Let's examine our eleventh example, y = log₁₀(3x + 9) + 1. This graph introduces a new transformation - addition inside the logarithm. How does this change the domain of the function compared to previous examples? Notice that the domain is now x > -3. Can you explain why? The addition of 1 outside the logarithm shifts the graph vertically. How does the combination of 3x + 9 inside the logarithm affect the graph's horizontal position? Can you identify where this graph intersects the y-axis? Remember, when x = 0, we're calculating log₁₀(9) + 1.

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