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Mathematics /AP Calculus AB: 6.3.1 The Exponential and Natural Log Functions

AP Calculus AB: 6.3.1 The Exponential and Natural Log Functions

Mathematics12 CardsCreated 8 months ago

This flashcard set introduces the exponential function 𝑒 𝑥 e x and the natural logarithm function ln 𝑥l nx, highlighting their properties, graphs, and inverse relationship. It also touches on their domains, asymptotes, and basic derivatives, offering conceptual and visual understanding of these foundational transcendental functions.

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The Exponential and Natural Log Functions

Raising the number e to the power x produces the exponential function. It is a positive, increasing function.
Taking the log to the base e of x (log e x) produces the natural log function, noted ln x. It is an increasing function defined only for positive x-values.
The exponential and natural log functions are inverses of each other.

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Key Terms

Term

Definition

The Exponential and Natural Log Functions

Raising the number e to the power x produces the exponential function. It is a positive, increasing function.
Taking the log...

note

On the left, the graph of the exponential function (e x ) shows that it is positive and increasing. The x-axis is a left horizontal asympto...

For which of these values is the natural log function not defined?

cosπ

Which of the following is the graph of y =2e^x+1?

This exponential graph is increasing and passes through the point (0, 3).

Which is the largest among the following expressions?

ln 10

If y(x)=e^ln(sinx+2), what is dy/dx?

cos x

Evaluate:

d/dx(e^2x+ln(2x))

2e^2x+1/x

Which of the following is the graph of

y = ln (2x) + 1, for x > 0?

This logarithmic graph is increasing and passes through the point (0.5, 1).

How many of the following expressions have the same value as 2? e^ln2, ln(e^2), ln2, 1 + ln e

3

Which is the smallest among the following expressions?

ln 3

Given the following expressions: e2, ln(1/2), ln(e), and sin 0, which of the following is the order of their values from the smallest to the largest?

ln(1/2), sin0, ln(e), e

Which of the following is equivalent to d/dx(e^cosx)?

−e^cosx sinx

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AP Calculus AB: 6.3.1 The Exponential and Natural Log Functions

Term	Definition
The Exponential and Natural Log Functions	Raising the number e to the power x produces the exponential function. It is a positive, increasing function. Taking the log to the base e of x (log e x) produces the natural log function, noted ln x. It is an increasing function defined only for positive x-values. The exponential and natural log functions are inverses of each other.
note	On the left, the graph of the exponential function (e x ) shows that it is positive and increasing. The x-axis is a left horizontal asymptote for the curve at. On the right, the graph of the natural log function (ln x) shows that it is increasing. It is only defined for positive x-values, in contrast to the exponential function. The y-axis is a vertical asymptote for the curve at. The exponential and natural log functions are inverses of each other. When you compose them in either order, they cancel each other out. Some mathematicians define ln x as the definite integral from one to x of the function 1/t. The derivative of e^x is e^x . The derivative of ln x is 1/x.
For which of these values is the natural log function not defined?	cosπ
Which of the following is the graph of y =2e^x+1?	This exponential graph is increasing and passes through the point (0, 3).
Which is the largest among the following expressions?	ln 10
If y(x)=e^ln(sinx+2), what is dy/dx?	cos x
Evaluate: d/dx(e^2x+ln(2x))	2e^2x+1/x
Which of the following is the graph of y = ln (2x) + 1, for x > 0?	This logarithmic graph is increasing and passes through the point (0.5, 1).
How many of the following expressions have the same value as 2? e^ln2, ln(e^2), ln2, 1 + ln e	3
Which is the smallest among the following expressions?	ln 3
Given the following expressions: e2, ln(1/2), ln(e), and sin 0, which of the following is the order of their values from the smallest to the largest?	ln(1/2), sin0, ln(e), e
Which of the following is equivalent to d/dx(e^cosx)?	−e^cosx sinx