Logarithms – making complex calculations easy
John Napier
John Wallis
Johann
Bernoulli
Jost Burgi
Introduction to Logarithmic Functions
GRAPHS OF EXPONENTIALS AND ITS INVERSE
•A logarithmic function is the inverse of an exponential function
•Exponential functions have the following characteristics:
Domain: {x є R}
Range: {y > 0}
Standard 11
Writing Exponential form to Logarithmic form
First we must learn how to read logarithmic form:
The expression log b y is read as “log of base b of y”
Examples:
log 5 125
log of base 5 of 125
log 6 36
log of base 6 of 36
1
log 3
5
log of base 3 of 1/5
Introduction to Logarithmic Functions
FINDING THE INVERSE OF AN EXPONENTIAL
x
y
a
yx = log
Inverse
Logarithmic
of the Exponential
Form
Exponential
Function
Function
Logarithms
Index
Power
Exponent
Logarithm
Base
102 = 100
Number
“10 raised to the power 2 gives 100”
“The power to which the base 10 must be raised to give 100 is 2”
“The logarithm to the base 10 of 100 is 2”
Log10100 = 2
Logarithms
y = bx
Logby = x
Logarithm
2
10
Base
= 100
Number
logby = x
is the inverse of
y = bx
Base
Logarithm
Log10100 = 2
Number
23 = 8
Log28 = 3
34 = 81
Log381 = 4
Log525 =2
52 = 25
Log93 = 1/2
91/2 = 3
Standard 11
Rewriting Logarithmic Equations
Exponential Form
2 1 
1
2
Logarithmic Form
log 2
1
 1
2
2 4  16
log 2 16  4
5 x  125
log 5 125  x
6 y  36
log 6 36  y
1
x
3 
9
1
log 3  x
9
103 = 1000
log101000 = 3
p = q2
logqp = 2
24 = 16
log216 = 4
xy = 2
logx2 = y
104 = 10,000
log1010000 = 4
pq = r
logpr = q
32 = 9
log39 = 2
logxy = z
xz = y
42 = 16
log416 = 2
loga5 = b
ab = 5
10-2 = 0.01
log100.01 = -2
logpq = r
pr = q
log464 = 3
43 = 64
c = logab
b = ac
log327 = 3
33 = 27
log366 = 1/2
361/2 = 6
log121= 0
120 = 1
103 = 1000
log101000 = 3
p = q2
logqp = 2
24 = 16
log216 = 4
xy = 2
logx2 = y
104 = 10,000
log1010000 = 4
pq = r
logpr = q
32 = 9
log39 = 2
logxy = z
xz = y
42 = 16
log416 = 2
loga5 = b
ab = 5
10-2 = 0.01
log100.01 = -2
logpq = r
pr = q
log464 = 3
43 = 64
c = logab
b = ac
log327 = 3
33 = 27
log366 = 1/2
361/2 = 6
log121= 0
120 = 1
103 = 1000
log101000 = 3
p = q2
logqp = 2
24 = 16
log216 = 4
xy = 2
logx2 = y
104 = 10,000
log1010000 = 4
pq = r
logpr = q
32 = 9
log39 = 2
logxy = z
xz = y
42 = 16
log416 = 2
loga5 = b
ab = 5
10-2 = 0.01
log100.01 = -2
logpq = r
pr = q
log464 = 3
43 = 64
c = logab
b = ac
log327 = 3
33 = 27
log366 = 1/2
361/2 = 6
log121= 0
120 = 1
103 = 1000
log101000 = 3
p = q2
logqp = 2
24 = 16
log216 = 4
xy = 2
logx2 = y
104 = 10,000
log1010000 = 4
pq = r
logpr = q
32 = 9
log39 = 2
logxy = z
xz = y
42 = 16
log416 = 2
loga5 = b
ab = 5
10-2 = 0.01
log100.01 = -2
logpq = r
pr = q
log464 = 3
43 = 64
c = logab
b = ac
log327 = 3
33 = 27
log366 = 1/2
361/2 = 6
log121= 0
120 = 1
103 = 1000
log101000 = 3
p = q2
logqp = 2
24 = 16
log216 = 4
xy = 2
logx2 = y
104 = 10,000
log1010000 = 4
pq = r
logpr = q
32 = 9
log39 = 2
logxy = z
xz = y
42 = 16
log416 = 2
loga5 = b
ab = 5
10-2 = 0.01
log100.01 = -2
logpq = r
pr = q
log464 = 3
43 = 64
c = logab
b = ac
log327 = 3
33 = 27
log366 = 1/2
361/2 = 6
log121= 0
120 = 1
Introduction to Logarithmic Functions
CHANGING FORMS
Example 1) Write the following into logarithmic form:
a) 33 = 27
b) 45 = 256
c) 27 = 128
d) (1/3)x=27
ANSWERS
Introduction to Logarithmic Functions
CHANGING FORMS
Example 1) Write the following into logarithmic form:
a) 33 = 27
log327=3
b) 45 = 256
log4256=5
c) 27 = 128
log2128=7
d) (1/3)x=27
log1/327=x
Standard 11
Simplifying Logarithmic Equations
Logarithmic Form
Exponential Form
Solution
log 4 16
4 x  16
x2
log 3 1
3y  1
y0
1
log 2
8
1
2 
8
z
z  3
log 4 2
4a  2
1
a
4
log 27 3
27  3
b
b
1
6
Introduction to Logarithmic Functions
CHANGING FORMS
Example 2) Write the following into exponential form:
a) log264=6
b) log255=1/2
c) log81=0
d) log1/31/9=2
ANSWERS
Introduction to Logarithmic Functions
CHANGING FORMS
Example 2) Write the following into exponential form:
a) log264=6
2^6=64
b) log255=1/2
25^1/2=5
c) log81=0
8^0=1
d) log1/31/9=2
1/3^2=1/9
Introduction to Logarithmic Functions
BASE 10 LOGS
Scientific calculators can perform logarithmic operations. Your calculator has
a LOG button.
This button represents logarithms in BASE 10 or log10
Example 4) Use your calculator to find the value of each of the following:
a) log101000
=3
b) log 50
= 1.699
c) log -1000
= Out of Domain