6.3.1 The Exponential and Natural Log Functions Flashcards
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
