Scientific Notations of Real Numbers and Logarithms
Express Large Numbers in Standard Form and Vice-versa
Let us suppose we are given 3 numbers: 2, 3 and 9.
Now, we know that 32 = 9
Also,
The above two expressions are formed by combining 2 and 3, and 2 and 9 respectively to get the third number.
Is there an expression wherein we can combine 3 and 9 to get 2?
3 and 9 can be combined to get 2 as:
Here, ‘log’ is the abbreviated form of a concept called ‘Logarithms’.
The expression can be read as ‘logarithm of 9 to the base 3 is equal to 2’.
In general, if a is any positive real number (except 1), n is any rational number such that , then n is called the logarithm of b to the base a, and is written as.
Thus, if and only if .
is called the exponential form and is called the logarithmic form.
The following are the properties of logarithms.
1. Since a is any positive real number (except 1), an is always a positive real number for every rational number n, i.e., b is always a positive real number.
Thus, logarithms are only defined for positive real numbers.
2. Since
Thus, and
where, a is any positive real number except 1
3. If
Then, and
⇒ x = y
Thus,
⇒ x = y
4. Logarithms to the base 10 are called common logarithms.
5. If no base is given, the base is always taken as 10.
For example, log 5 = log10 5
6. Logarithms to the base e, where e is an irrational number lying between 2 and 3 are called natural logarithms. The approximate value of e correct up to 1 decimal place is 2.7 and is given by
Natural logarithm of 5 can be written as:
loge5 = ln 5
Let us consider the following example.
Convert the following into logarithmic form.
(i) 53 = 125
(ii)
There are three standard laws of logarithms.
(i) Product Law
In general,
(ii) Quotient Law
(iii) Power Law
On the basis of the above laws, we have
For a and b two positive numbers, .
Also, we know that, log of a number at the same base is 1 i.e .
Example 1:
Solve for x.
(i) log7 343 = 5x − 4
(ii) logx 216 = 3
Solution:
(i)
(ii)
Example 2:
If what is x?
Solution:
Now,
Example: 3
Solve for x.
(i)
(ii)
Solution:
(i)
We know that,
For a and b two positive numbers, .
Therefore,
(ii)
We know that, log of a number at the same base is 1 i.e .
Therefore,
We know:
101 = 10 and 102 = 100
∴ log10 10 = 1 and log10 100 = 2 i.e. log10 10p = p .....(1)
Also, for 1 < N < 10, 0 < log10 N < 1. .....(2)
Characteristic of Logarithm
Every…
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