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 3^{2} = 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), *a*^{n} 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 = log_{10} 5

Let us consider the following example.

Convert the following into logarithmic form.

(i) 5^{3} = 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, ${\mathrm{log}}_{b}a=\frac{1}{{\mathrm{log}}_{a}b}$.

$\Rightarrow {\mathrm{log}}_{b}a\times {\mathrm{log}}_{a}b=1$

Also, we know that, log of a number at the same base is 1 i.e ${\mathrm{log}}_{a}a=1$.

$\Rightarrow x{\mathrm{log}}_{a}a=x\phantom{\rule{0ex}{0ex}}\Rightarrow {\mathrm{log}}_{a}{a}^{x}=x$

**Example 1:**

Solve for *x*.

(i) log_{7} 343 = 5*x* − 4

(ii) log_{x} 216 = 3

**Solution: **

(i)

(ii)

**Example 2:**

If what is *x*?

**Solution:**

Now,

**Example: 3**

Solve for *x*.

(i) ${\mathrm{log}}_{13}7\times {\mathrm{log}}_{7}13=x$

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

**Solution:**

(i)

We know that,

For *a* and *b* two positive numbers, ${\mathrm{log}}_{b}a=\frac{1}{{\mathrm{log}}_{a}b}$.

$\Rightarrow {\mathrm{log}}_{b}a\times {\mathrm{log}}_{a}b=1$

Therefore,

${\mathrm{log}}_{13}7\times {\mathrm{log}}_{7}13=x=1\phantom{\rule{0ex}{0ex}}\Rightarrow x=1$

(ii)

We know that, log of a number at the same base is 1 i.e ${\mathrm{log}}_{a}a=1$.

$\Rightarrow x{\mathrm{log}}_{a}a=x\phantom{\rule{0ex}{0ex}}\Rightarrow {\mathrm{log}}_{a}{a}^{x}=x$

Therefore,

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

**Common Logarithms**

We know:

10

^{1}= 10 and 10

^{2}= 100

∴ log

_{10}

_{ }10 = 1 and log

_{10}100 = 2

For 10 <

*n*< 100, 1 < log

_{10}

*n*< 2.

Example: log

_{10}20 = 1.3010 and log

_{10}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*,

log

_{10}

*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 log

_{10}

*N*=

*m*− 1…

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