Tangent Lines and Logarithmic / Exponential Functions

Tangent Line and Logarithmic/Exponential Functions

This project is going to consider the relationship between a point (a, b) on the natural logarithmic and exponential functions and its relationship to the intercept of the tangent line to the respective functions at the point (a, b).

Hint: Recall the distance between two values a, b is
I. Natural Logarithm Function
1. Let L be the tangent line to at the point (1, 0) [See Figure 1]:
a. Find the equation of the line, L:
b. Let c be the y - intercept of the line L: What is the distance between c and 0?
2. Now, let T be the tangent line to at the point (a, b), where a, b are real numbers:
a. Find the equation of the line, T:
b. Let c be the y - intercept of the line T: What is the distance between c and b?
II. Exponential Function
3. Let L be the tangent line to at the point (0, 1) [See Figure 2]:
a. Find the equation of the line, L:
b. Let c be the x - intercept of the line L: What is the distance between c and 0?
4. Now, let T be the tangent line to at the point (a, b) where a, b are real numbers (Hint: ):
a. Find the equation of the line, T:
b. Let c be the x - intercept of the line T:
What is the distance between c and a?

III. Analysis:
In Part I of this project, you found that at any point (a, b) on the graph of , the distance between b (the y - coordinate) and the y - intercept of the tangent line to (a, b) is ____________.

In Part II of this project, you found that at any point (a, b) on the graph of , the distance between a (the x - coordinate) and the x - intercept of the tangent line to (a, b) is ____________.

In your own words, explain why these distances are the same.
(Hint: and share a special relationship)