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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, 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, logba=1logab. ⇒logba ×logab=1 Also, we know that, log of a number at the same base is 1 i.e logaa=1. ⇒xlogaa=x⇒logaax=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) log137 ×log713=x (ii) log559=x Solution: (i) We know that, For a and b two positive numbers, logba=1logab. ⇒logba ×logab=1 Therefore, log137 ×log713=x=1⇒x=1 (ii) We know that, log of a number at the same base is 1 i.e logaa=1. ⇒xlogaa=x⇒logaax=x Therefore, log559=x=9⇒x=9

Let us suppose we are given 3 numbers: 2, 3 and 9.

Now, we know that 32 = 9

Also, 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, logba=1logab. ⇒logba ×logab=1 Also, we know that, log of a number at the same base is 1 i.e logaa=1. ⇒xlogaa=x⇒logaax=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) log137 ×log713=x (ii) log559=x Solution: (i) We know that, For a and b two positive numbers, logba=1logab. ⇒logba ×logab=1 Therefore, log137 ×log713=x=1⇒x=1 (ii) We know that, log of a number at the same base is 1 i.e logaa=1. ⇒xlogaa=x⇒logaax=x Therefore, log559=x=9⇒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. The characteristic of the logarithm of a positive number 0 < N < 1 is negative. If there are k zeroes immediately after the decimal point, then the characteristic of the logarithm is log10 N = −(k + 1). Example: Characteristic of log10 1234 = 4 − 1 = 3 because 1234 has 4 digits in its integral part. Characteristic of log10 123.4 = 3 − 1 = 2 because 123.4 has 3 digits in its integral part. Characteristic of log10 0.4 = −(0 + 1) = −1 because there are no zeroes immediately after the decimal point. Characteristic of log10 0.0045 = −(2 + 1) = −3 because there are 2 zeroes immediately after the decimal point. Mantissa of Logarithm We use l…

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