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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, $\sqrt{9}=3$

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

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, ${\mathrm{log}}_{b}a=\frac{1}{{\mathrm{log}}_{a}b}$.

Also, we know that, log of a number at the same base is 1 i.e ${\mathrm{log}}_{a}a=1$.
$⇒x{\mathrm{log}}_{a}a=x\phantom{\rule{0ex}{0ex}}⇒{\mathrm{log}}_{a}{a}^{x}=x$

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)
${\mathrm{log}}_{5}{5}^{9}=x$

Solution:

(i)
We know that,
For a and b two positive numbers, ${\mathrm{log}}_{b}a=\frac{1}{{\mathrm{log}}_{a}b}$.

Therefore,

(ii)
We know that, log of a number at the same base is 1 i.e ${\mathrm{log}}_{a}a=1$.
$⇒x{\mathrm{log}}_{a}a=x\phantom{\rule{0ex}{0ex}}⇒{\mathrm{log}}_{a}{a}^{x}=x$
Therefore,

${\mathrm{log}}_{5}{5}^{9}=x=9\phantom{\rule{0ex}{0ex}}⇒x=9$

Common Logarithms

We know:

101 = 10 and 102 = 100

∴ log10 10 = 1 and log10 100 = 2

For 10 < n < 100, 1 < log10 n < 2.

Example: log10 20 = 1.3010 and log10 25 = 1.3979

Here, the integral part 1 is called the characteristic of the logarithm and the fractional part is called the mantissa of the logarithm.

For any positive number N,

log10 N = Characteristic + Mantissa

Characteristic of Logarithm

The characteristic of the logarithm of a positive number N ≥ 1 is positive.
If a positive number N ≥ 1 has m digits in its integral part, then subtract 1 from  m to find the characteristic of the logarithm—that is, characteristic log10 N = m − 1…

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