Vertical asymptote of natural log | Limits | Differential Calculus | Khan Academy

TL;DR

The content explains the domain and plot of the natural logarithm function, highlighting the importance of taking the natural log of positive values and identifying the vertical asymptote at x = 3.

Transcript

• [Narrator] Right over here, we've defined y as a function of x, where y is equal to the natural log of x minus three. What I encourage you to do right now is to pause this video and think about for what x values is this function actually defined? Or another way of thinking about it, what is the domain of this function, and then try to plot this f... Read More

Key Insights

• ☺️ The natural logarithm function is defined for x values greater than 3.
• 👈 The function intersects the x-axis at the point (4, 0).
• 👉 As x approaches 3 from the right, the function rapidly decreases towards negative infinity.
• ☺️ The function is undefined for x values less than or equal to 3.

Questions & Answers

Q: What is the domain of the natural logarithm function in this case?

The function is defined for x values greater than 3, so the domain can be expressed as x > 3 or the set of all real numbers greater than 3.

Q: How can we find the point where the function equals zero?

By raising e to the natural log of x minus three, we find that x equals four gives the function a value of zero. So, the point (4, 0) is where the function intersects the x-axis.

Q: What happens to the function as x approaches 3 from the right?

As x gets closer to 3 from values greater than 3, the function takes increasingly negative values. This indicates a vertical asymptote at x = 3, where the function approaches negative infinity.

Q: Why is it important for the input of the natural log function to be positive?

The natural logarithm function is only defined for positive inputs. Taking the natural log of a negative or zero value is undefined in this context. Therefore, the function is only valid for positive values of x.

Summary & Key Takeaways

• The content discusses the domain of the natural logarithm function, which is defined for x greater than 3.

• It explains how to plot the function by finding key points, such as when the function equals zero.

• The video demonstrates that as x approaches 3 from the right side, the function rapidly decreases towards negative infinity, indicating a vertical asymptote at x = 3.