The logarithmic Functions

1
The logarithmic Functions
2
Revision
We have learnt that
x by
ltgt
y logb x
ltgt means if and only if.
3
Two important Results
4
Laws of Logarithms
If M,N and b are positive real numbers, b ? 1,
and n is any real number, then
5
Class Practice
Example 1 Without using calculators, find (a)
log10 50 - log10 2 2 log10 2
6
Important formula
We have stated that
Let y loga M.
In logarithmic form with base 10, we have
y log10 a
y log10 a
7
Important base, e
We have learnt that
y ln x
8
Two important results
Change of base formula
9
Class Practice
Example Simplify the followings
3
ax
x4
x4
x5
5
0.5
1/3
10
Class Practice
Example By using calculators, find
(a) log3 7 (b) log7 0.89
11
Example 1
As long as a plant or animal is alive, carbon-14
is maintained at a constant level in its tissues.
Once dead, however, it ceases taking in carbon
and the carbon-14 diminishes by radioactive decay
according to the equation where t is time in
years. Estimate the age of a skull uncovered at
an archaeological site if 10 of the original
amount of carbon-14 was still present.
12
Example 1
13
Example 1
Ao is the original amount of carbon-14.
Now, 10 of the original amount of carbon-14 was
still present.
0.1 Ao
14
Example 1
Taking natural logarithm,
Therefore, the age is 18569 years.
15
Example 2
In a certain bacterial culture there are 420 000
bacteria at the end of two days and 565 0000
present at the end of four days. Assuming that
the bacterial population follows the law of
exponential growth, find (a) the number present
at the beginning , (b) the number present after
one day, (c ) the number of days required for
there to be 420 000 00 bacteria.
16
Example 2
Since, the bacterial population follows the law
of exponential growth.
When t 2, Q(2) 420 000
When t 4, Q(4) 565 0000