Introduction:
Exponential function can be represented as f(x) = bx where b is real number and x is a variable. Domain of any function f(x) is a set of all real values on X-axis for which function is defined and exponential function is defined for all real value of x on X-axis so domain of exponential function is all real number. For function f(x) = bx value of f(x) will be always positive if b is positive and always negative if b is negative so in range of function there will no negative number if b is positive and there will be no positive number in range of function f(x) if a is negative. Exponential functions will be strictly increasing or strictly decreasing depending on value of b. If b >1 then function will be strictly increasing and if b < 1 function will be strictly decreasing for example function y=5x is strictly increasing and function y = (1/5) x is strictly decreasing function. Inverse of exponential function is logarithmic function, logarithmic function can be written as y = logxy where x and y both are positive numbers. Exponential function can also be written as y =ex where e is a real number.
More on Exponential Functions:
For exponential function f(x) = bx , b is called base of exponential function. If two equal exponential functions have same base then their power will be also same for example if bx = by then we can write as x = y, to find inverse of exponential function just inverse the position of x and y in function.Looking out for more help on college algebra solver in algebra by visiting listed websites.
Basic properties of exponential functions:
`e^x e^y = e^(x+y)`
`(e^x)^p = e^(px)`
`de^x/dx = e^x`
`de^(ax)/dx = ae^(ax)`
`d^n e^(ax)/dx^n = a^n e^(ax)`
`e^x/e^y = e^(x-y)`
Example Problem:
Example 1: What will be value of x if 2x = 8?
Solution: 2x = 8
2x = 23
Base is same for both side of equation so
x = 3
Example 2: Find inverse of exponential function y = 5x
Solution: Given function is y = 5x
To find inverse of this function substitute y for x and x for y
x = 5y
log5 x = log5 5y
= y log5 5
= y
So, Inverse of given function is y = log5 x
Example 3: Find the exponential function y = 7x and also graph the function.
Solution: A set of ordered pairs for the equation is shown below.
Putting the value x = -0.5 we get
`y = 7^-0.5`
= 0.3779
(x,y) = (-0.5, 0.3779)
Putting the value x = -0.25 we get
`y = 7^-0.25`
= 0.614
(x,y) = (-0.25, 0.614)
Putting the value x = 0 we get
`y = 7^0`
= 1
(x,y) = (0,1)
Putting the value x = 0.25 we get
`y = 7^0.25`
= 1.626
(x,y) = (0.25, 1.626)
Plotting the given points to get the graph of the exponential function,
This graph shows the part of the exponential function between the co-ordinate points. So, it might not look like a curve but if we plot all the points over the real domain, we get the exact curve.
Exponential function can be represented as f(x) = bx where b is real number and x is a variable. Domain of any function f(x) is a set of all real values on X-axis for which function is defined and exponential function is defined for all real value of x on X-axis so domain of exponential function is all real number. For function f(x) = bx value of f(x) will be always positive if b is positive and always negative if b is negative so in range of function there will no negative number if b is positive and there will be no positive number in range of function f(x) if a is negative. Exponential functions will be strictly increasing or strictly decreasing depending on value of b. If b >1 then function will be strictly increasing and if b < 1 function will be strictly decreasing for example function y=5x is strictly increasing and function y = (1/5) x is strictly decreasing function. Inverse of exponential function is logarithmic function, logarithmic function can be written as y = logxy where x and y both are positive numbers. Exponential function can also be written as y =ex where e is a real number.
More on Exponential Functions:
For exponential function f(x) = bx , b is called base of exponential function. If two equal exponential functions have same base then their power will be also same for example if bx = by then we can write as x = y, to find inverse of exponential function just inverse the position of x and y in function.Looking out for more help on college algebra solver in algebra by visiting listed websites.
Basic properties of exponential functions:
`e^x e^y = e^(x+y)`
`(e^x)^p = e^(px)`
`de^x/dx = e^x`
`de^(ax)/dx = ae^(ax)`
`d^n e^(ax)/dx^n = a^n e^(ax)`
`e^x/e^y = e^(x-y)`
Example Problem:
Example 1: What will be value of x if 2x = 8?
Solution: 2x = 8
2x = 23
Base is same for both side of equation so
x = 3
Example 2: Find inverse of exponential function y = 5x
Solution: Given function is y = 5x
To find inverse of this function substitute y for x and x for y
x = 5y
log5 x = log5 5y
= y log5 5
= y
So, Inverse of given function is y = log5 x
Example 3: Find the exponential function y = 7x and also graph the function.
Solution: A set of ordered pairs for the equation is shown below.
Putting the value x = -0.5 we get
`y = 7^-0.5`
= 0.3779
(x,y) = (-0.5, 0.3779)
Putting the value x = -0.25 we get
`y = 7^-0.25`
= 0.614
(x,y) = (-0.25, 0.614)
Putting the value x = 0 we get
`y = 7^0`
= 1
(x,y) = (0,1)
Putting the value x = 0.25 we get
`y = 7^0.25`
= 1.626
(x,y) = (0.25, 1.626)
Plotting the given points to get the graph of the exponential function,
This graph shows the part of the exponential function between the co-ordinate points. So, it might not look like a curve but if we plot all the points over the real domain, we get the exact curve.
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