Functions
Exponential and logarithmic functions
Exponential Functions Logarithmic Functions
Exponential Functions
The function represented by f(x)= ex
is called an exponential function with base e and exponent x. The domain of f is the set of all real numbers
which is ] – , [
Laws of Exponents ď ľ Lets start by these rules:
Limits
Derivative ď ľ Derivate of an exponential function:
Table of Variation Derivate of an exponential function: f(x)= ex then f’(x)= ex>0 f’(x)>0 then f(x) is strictly increasing
Graph ď ľ Sketch the graph of the exponential function f(x) = ex.
Solution First, recall that the domain of this function is the set of real numbers Next, putting x = 0 gives y = e0 = 1, which is the y-intercept. Y- intercept means the point where the graph cuts y-axis. So here we have the point (0,1) would be the intersection between the curve and y-axis. There is no x-intercept, there is no value of x for which y = 0, here we have to note that ex >0. This means ex is always positive so it is always above x-axis.
Graph From the limit of ex at -∞ which is zero, we conclude that
there is a horizontal asymptote at y = 0 (x-axis). Furthermore, ex increases without bound when x increases since f’(x)= ex >0 From the limit of ex at +∞ which is +∞, we conclude the range of f is the interval ]0, [.
Graph Sketch the graph of the exponential function f(x) = ex.
Solution
y 4
f (x ) = e x
e
–e
e
x
Summary of Exponential Functions The exponential function y = ex has the following
properties: 1. Its domain is ]– , [. 2. Its range is ]0, [. 3. Its graph passes through the point (0, 1) 4. It is continuous on ]– , [. 5. It is increasing on ]– , [
Exercise Sketch the graph of the exponential function f(x) = e–x.
Solution: Please follow the same steps, you should get the below curve. Sketching the graph: y 5
3
1 –3
–1
f(x) = e–x 1
3
x
Could you do it? If you could perform all steps and get a correct curve, you
can proceed to logarithmic functions. If not, please review the lesson. If you need help, please contact us and we will provide more
resources and explanation to make it easier for you: ibrahim.dhaini.2014@gmail.com
Logarithmic Functions
Logarithms We’ve discussed exponential equations of the form
y = ex If we want to solve the above equation to get the value of x,
then we are searching for the logarithm of y.
✦ Logarithm of x to the base e y = ln (x) if and only if x = ey
(x > 0)
Examples ď ľ Solve ln x = 4 :
Solution ď ľ By definition, ln x = 4 implies x = e4 .
Logarithmic Notation
log x = log10 x ln x = loge x
Common logarithm Natural logarithm
Laws of Logarithms
Laws of Logarithms
Logarithmic Function ď ľ The function defined by f(x)= ln x
is called the logarithmic function. ď ľ The domain of f is the set of all positive numbers, that means we are allowed to use only positive values of x
Properties of Logarithmic Functions
The logarithmic function
y = ln x has the following properties: 1. Its domain is ]0, [. 2. Its range is ]– , [. 3. Its graph passes through the point (1, 0). 4. It is continuous on ]0, [. 5. It is increasing on ]0, [
Graph Sketch the graph of the function y = ln x.
Solution We first sketch the graph of y = ex. The required graph is the mirror image of the y graph of y = ex with respect to the line y = x:
y = ex
y=x
y = ln x 1 1
x
Graph Lets use the properties of f(x)= ln x to sketch the graph. Allowed values of x are from 0 to +∞ (x>0) This function is strictly increasing ( f’(x) = 1/x >0) Ln 1 = 0 Limit at 0 is -∞ so x=0 is a vertical asymptote Limit at +∞ is +∞
Graph
Properties Relating Exponential and Logarithmic Functions
ď ľ Properties relating ex and ln x:
eln x = x ln ex = x
(x > 0) (for any real number x)
Equations with ln x Solve the equation 5 ln x + 3 = 0.
Solution Add – 3 to both sides of the equation and then divide both sides of the equation by 5 to obtain: 5ln x = -3 3 ln x = - = -0.6 5 and so: eln x = e -0.6 x = e -0.6 x » 0.55
Examples ď ľ Expand and simplify the expression:
x2 x2 - 1 x 2 ( x 2 - 1)1/2 ln = ln x e ex = ln x 2 + ln( x 2 - 1)1/2 - ln e x 1 = 2 ln x + ln( x 2 - 1) - x ln e 2 1 = 2 ln x + ln( x 2 - 1) - x 2
Still need some exercises? ď ľ In the attached presentation, you can find some exercises
to help you solve an equation with ln(x) and ex . Microsoft PowerPoint 97-2003 Presentation
Microsoft PowerPoint 97-2003 Presentation
End of ChaptE r