This document provides an overview of exponential and logarithmic functions. It defines one-to-one functions and inverse functions. It explains how to find the inverse of a one-to-one function and shows that the inverse of f(x) is f-1(x). Properties of exponential functions like f(x)=ax and logarithmic functions like f(x)=logax are described. The product, quotient, and power rules for logarithms are outlined along with examples. Finally, it discusses how to solve exponential and logarithmic equations using properties of these functions.

1. Exponential and
Logarithmic Functions
Presenter: Njabulo “Mr-N” Nkabinde
Grade 12

2. Content

3. One-to-One Functions
• A function is a one-to-one function if each value in the range corresponds with
exactly one value in the domain.
• For a function to be one-to-one, it must not only pass the vertical line test, but
also the horizontal line test.
y
y
x
Function
x
Not a one-to-one
function
y
x
One-to one
function

4. Inverse Functions
If f(x) is a one-to-one function with ordered pairs of
the form (x,y), its inverse function, f -1(x), is a oneto-one function with ordered pairs of the form (y,x).
Function:
Inverse Function:
{(2, 6), (5,4), (0, 12), (4, 1)}
{(6, 2), (4,5), (12, 0), (1, 4)}
• Only one-to-one functions have inverse functions.
• Note that the domain of the function becomes the
range of the inverse function, and the range
becomes the domain of the inverse function.

5. To Find the Inverse Function of a One-to-One Function
1. Replace f(x) with y.
2. Interchange the two variables x and y.
3. Solve the equation for y.
4. Replace y with f –1(x). (This gives the inverse function
using inverse function notation.)
Example:
Find the inverse function of f  x   x  1, x  1.
Graph f(x) and f(x) –1 on the same axes.

6. f  x 
x  1,  1
y
x 1
Replace f(x) with y.
x
y 1
Interchange x and y.
x 
2

y 1

2
Solve for y.
x2  y  1
x2  1  y
f 1( x)  x 2  1, x  0
Replace y with f –1(x) .

7. f 1( x)  x 2  1, x  0
f  x   x  1, x  1
Note that the symmetry is about the line y = x.

8. If two functions f(x) and f –1(x) are inverses of each
other,
.
( f f 1)( x)  x and ( f 1 f )( x)  x
Example:
f  x   x  1, x  1 and f 1( x)  x 2  1, x  0
Show that ( f
f  x 
(f
1
f 1)( x)  x and ( f 1 f )( x)  x..
x 1
f )( x)  x  1  1
2
 x2  x
f 1( x)  x 2  1
1
f ( x) 


2
x 1 1
 x 1 1  x

10. For any real number a > 0 and a  1,
f(x) = ax
is an exponential function.
For all exponential functions of this form,
1. The domain of the function is (, ).
2. The range of the function is (0, ).
3. The graph passes through the points
(1, 1 ),  0,1 , 1, a  .
a

11. Example:
Graph the function f(x) = 3x.
1, 3 
 -1, 1 
3
 0,1
Domain: (-∞,∞)
Range: {y|y > 0}

12. Example:

x
Graph the function f(x) = 1 .
3
Notice that each
graph passes
through the
point (0, 1).
 -1, 3
 0,1
 1, 1 
3
Domain:
Range: {y|y > 0}

13. Logarithmic Functions
For all positive numbers a, where a  1,
y = logax means x = ay.
logarithm
(exponent)
exponent
number
y = logax
base
x = ay
means
number
base

14. Exponential Form
Logarithmic Form
50 = 1
log101= 0
23 = 8
log28= 3
4
1
  = 16
log1 2 1 = 4
16
6-2 = 1
36
log6 1 = -2
36
1
2

15. For all logarithmic functions of the form y = logax or
f(x) = logax, where a > 0, a  1, and x > 0,
1. The domain of the function is (0, ) .
2. The range of the function is (, ) .
3. The graph passes through the points
( 1 , 1), 1,0  ,  a,1 .
a

16. Logarithmic Graphs
Example:
Graph the function f(x) = log10x.
10,1
1, 0 
1
 10 , -1
Notice that the
graph passes
through the
point (1,0).
Domain: {x|x > 0}
Range:

17. Exponential Function
Logarithmic Function
y = ax (a > 0, a  1)
y = logax (a > 0, a  1)
Domain:
Range:
Points on
Graph:
 0,  
 ,  
 ,  
 0,  
1
 a , 1
1
 1, a 
 0,1
1, a 
x becomes y
y becomes x
1,0
 a,1

18. Exponential vs.
Logarithmic Graphs
f(x)
f(x) = 10x
f(x) = log10x
f -1(x)
Notice that the two graphs are inverse functions.

20. Product Rule for Logarithms
For positive real numbers x, y, and a, a  1,
loga xy  loga x  loga y
Example:
log5(4 · 7) = log54 + log57
log10(100 · 1000) = log10100 + log101000 = 2 + 3 = 5

21. Quotient Rule for Logarithms
For positive real numbers x, y, and a, a  1,
log a x  log a x  log a y
y
Property 1
Example:
log 7 10  log 7 10  log 7 2
2
log10 1  log10 1  log10 1000  0  3  3
1000

22. Power Rule for Logarithms
If x and y are positive real numbers, a  1, and n is any
real number, then
log a x n  n log a x
Property 2
Example:
log 9 34  4 log 9 3
log10 1002  2 log10 100  2  2  4

23. Additional Properties of Logarithms
If a > 0, and a  1,
log a a x  x
aloga x  x (x  0)
Example:
log 9 9 4  4
log10 106  6
Property 4
Property 5

24. Example:
Write the following as the logarithm of a single
expression.
5log6(x  3)  [2log 6(x  4)  3log 6 x] 
log 6(x  3)5  [log 6(x  4) 2  log 6 x3] 
Power Rule
log 6(x  3)5  [log 6(x  4) 2  x3] 
Product Rule
 (x  3)5 
log 6 
(x  4)2 x3 


Quotient Rule

26. Properties for Solving Exponential and
Logarithmic Equations
a. If x = y, ax = ay.
Properties 6a-6d
b. If ax = ay, then x = y.
c. If x = y, then logbx = logby (x > 0, y > 0).
d. If logbx = logby, then x = y (x > 0, y > 0).

27. Solving Equations
Example:
Solve the equation 4 x  256.
2 
2
x
 28
Rewrite each side with the same base.
22 x  28
2x  8
Property 6b.
x4
Solve for x.

28. Solving Equations
Example:
Solve the equation log(x  3)  log x  log 4.
log(x  3)x  log 4
Product Rule
(x  3)x  4
Property 6d.
x 2  3x  4
x 2  3x  4  0
Check:
log(4  3)  log(4)  log 4.
Stop! Logs of negative numbers
are not real numbers.
(x  4)(x  1)  0
log(1  3)  log(1)  log 4.
x  4 or x  1
log 4  0  log 4
log 4  log 4 True

29. Reference
http://www.slideshare.net/leingang
http://www.slideshare.net/hisema01?utm_campaign=profiletracking&utm_m
edium=sssite&utm_source=ssslideview
http://www.slideshare.net/jessicagarcia62?utm_campaign=profiletracking&ut
m_medium=sssite&utm_source=ssslideview