BASIC PROPERTIES OF LOGARITHMIC FUNCTIONS
If a is any positive real number which in not equal to 1, then a function f (x) defined as f (x) = 
for x> 0 is called a logarithmic function.
Examples:
i. 
ii. 
iii. 
iv. 
Graphs of logarithmic functions
Draw graph of f (x) = 
Solution
Properties of logarithmic functions
i. It’s graph passes through (1,0)
ii. It’s domain is ( x : x > 0)
iii.It’s range is (y:y∑IR)
iv.It’s a one to one function
v. It is the inverse of exponential function
vi.It’s vertical asymptote is at x = 0
vii.No horizontal asymptote
LAWS OF LOGARITHMS:
= 
+ 
ii).
= 
– 
= n 
= n
= 0
= 1
=x , x > 0
CHANGING BASES OF LOGARITHM
Let y = 
Change it into base b
Steps:
i. Express logarithmic function into exponential function
Y = 
…………. (i)
N = 
……………. (ii)
Introduce logarithm to base b both sides:
N = 
= 
= y 
………….. (iii)
Divide (iii) by 
………… (iii)
y = 
Examples
Evaluate the following to four decimal places
1. 
2. 
Solution
1. 
Let y = 
= 5.2
Introducing natural logarithm
In 
= In 5.2
yln 3 = in 5.2
= 
= 
y = 1.5007
2. 
Let y = 
= 0.0372
In 
= ln 0.0372
= 
= 
y = 4.7488
EXERCISE
1. Evaluate the following correct to 4 decimal places
a) Log 49236
b) Ln 54.02
c) 
d) 
e) 
f) 
2. Change to base e
a) Log 5.3147
b) Log 0.053
3. Change to base 10
a) ln2.0103
b) ln 47.486
4. Draw graphs of
a) f (x) = 
b) f (x) = 
INTEGRATION OF TRIGONOMETRIC FUNCTIONS
a) Integration of sin x
Recalling;
(
) = -k
= 
Example
Compute
Solution
Let u = 5x
Du = 5dx
dx = 
= 
= 
=
b) Integration of 
Recalling
= k 
= 
c) Integration of sec2x
Recalling
= k sec2kx
ʃsec2kxdx = 
