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

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 $e = 1 +\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+.....$ 
Natural logarithm of 5 can be written as:
log_e5 =  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, ${\log }_{b}a=\frac{1}{{\log }_{a}b}$.
$\Rightarrow {\log }_{b}a \times {\log }_{a}b=1$

Also, we know that, log of a number at the same base is 1 i.e ${\log }_{a}a=1$.
$$\begin{gathered} \Rightarrow x{\log }_{a}a=x \\ \Rightarrow {\log }_a{a}^x=x \end{gathered}$$

Example 1:

Solve for x.

(i) log₇ 343 = 5x − 4

(ii) log_x 216 = 3

Solution: 

(i)

(ii)

Example 2:

If   what is x?

Solution:

Now,

Example: 3

Solve for x.

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

(ii) ${\log }_5{5}^9=x$

Solution:

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

Therefore,

$$\begin{gathered} {\log }_{13}7 \times {\log }_{7}13=x=1 \\ \Rightarrow x=1 \end{gathered}$$


(ii)
We know that, log of a number at the same base is 1 i.e ${\log }_{a}a=1$.
$$\begin{gathered} \Rightarrow x{\log }_{a}a=x \\ \Rightarrow {\log }_a{a}^x=x \end{gathered}$$
Therefore,

$$\begin{gathered} {\log }_5{5}^9=x=9 \\ \Rightarrow x=9 \end{gathered}$$
 

Common Logarithms

We know:

10¹ = 10 and 10² = 100

∴ log₁₀ 10 = 1 and log₁₀ 100 = 2  i.e.  log₁₀ 10^p = p           .....(1)

Also, for 1 < N < 10, 0 < log₁₀ N < 1.                                 .....(2)

Characteristic of Logarithm

Every…