Functions Exponential and logarithmic functions u Exponential Functions
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Functions Exponential and logarithmic functions u Exponential Functions u Logarithmic Functions
Exponential Functions u The function represented by f(x)= ex is called an exponential function with base e and exponent x. u The domain of f is the set of all real numbers which is ] – , [
Laws of Exponents u Lets start by these rules:
Limits
Derivative u Derivate of an exponential function:
Table of Variation u Derivate of an exponential function: u f(x)= ex then f’(x)= ex>0 u f’(x)>0 then f(x) is strictly increasing
Graph u 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 = e 0 = 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 u From the limit of ex at -∞ which is zero, we conclude that there is a horizontal asymptote at y = 0 (x-axis). u Furthermore, ex increases without bound when x increases since f’(x)= ex >0 u From the limit of ex at +∞ which is +∞, we conclude the range of f is the interval ]0, [.
Graph u 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 u 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 u Sketch the graph of the exponential function f(x) = e–x. Solution: Please follow the same steps, you should get the below curve. u Sketching the graph: y 5 3 1 – 3 – 1 f(x) = e–x 1 3 x
Could you do it? u If you could perform all steps and get a correct curve, you can proceed to logarithmic functions. u If not, please review the lesson. u 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 u We’ve discussed exponential equations of the form y = ex u 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 u Solve ln x = 4 : Solution u By definition, ln x = 4 implies x = e 4.
Logarithmic Notation log x = log 10 x ln x = loge x Common logarithm Natural logarithm
Laws of Logarithms
Laws of Logarithms
Logarithmic Function u The function defined by f(x)= ln x is called the logarithmic function. u 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 u 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 u Sketch the graph of the function y = ln x. Solution u We first sketch the graph of y = ex. u The required graph is the mirror image of the y x graph of y = e with respect to the line y = x: y = ex y=x y = ln x 1 1 x
Graph u Lets use the properties of f(x)= ln x to sketch the graph. u Allowed values of x are from 0 to +∞ (x>0) u This function is strictly increasing ( f’(x) = 1/x >0) u Ln 1 = 0 u Limit at 0 is -∞ so x=0 is a vertical asymptote u Limit at +∞ is +∞
Graph
Properties Relating Exponential and Logarithmic Functions u Properties relating ex and ln x: eln x = x ln ex = x (x > 0) (for any real number x)
Equations with ln x u Solve the equation 5 ln x + 3 = 0. Solution u Add – 3 to both sides of the equation and then divide both sides of the equation by 5 to obtain: and so:
Examples u Expand simplify the expression:
Still need some exercises? u In the attached presentation, you can find some exercises to help you solve an equation with ln(x) and ex.
End of Chapter
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