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Logarithms

The inventor of logarithms is John Napier, and was born in Scotland and followed mathematics as a hobby. With the help of logarithm, we can abbreviate the process of multiplication and division.

Logarithms are used to represent exponential notation in a different way which enables us to perform complex mathematical calculations easily. The logarithm of a number to a given base is equal to the exponent to which the base must be raised in order to obtain that number. In the expression log (a) b = c , 'a' raised to the power of 'c' will give you the result 'b'. Thus, the two expressions log (a) b = c and a^c = b essentially represent the same statement. Below you could see how to simplify logarithms.

Logarithm Definition: If three numbers m, n and x are related such that `m^n` `=x`

then n is called the logarithm of the number 'x' to the base 'm' and is written as, log_m x = n

What is a Logarithm

Let us take 2 ⁵= 32, what power of 2 gives 32? The answer is 5

So the 5 is termed as exponent and 2 is termed as base

Means the exponent 5 is the logarithm of 32 to the base 2.

If 4²= 16, then logarithm of 16 to the base 4 is equal to 2 ( log ₄ 16 = 2)

4²= 16 is called as exponential form

(log ₄ 16 = 2) is called as logarithm form.

Properties of Logarithms

1.Logarithm of unity to any base(`!=` 1) is zero.

log_a1 = 0 ( since `a^0` =1)

1. log ₁₀1 = 0

2. log ₅1 = 0

2. Logarithm of positive number to the same base is equal to 1

log _a a = 1 ( since `a^1` =a)

1. log₅ 5 = 1

2. log ₁₀ 10 = 1

It is written as antilog₂ 5 = 32

Hence, 2⁵ = 32 `hArr` log₂ 32 = 5 `hArr` antilog₂5 = 32

5⁴ = 625 `hArr` log₅ 625 = 4 `hArr` antilog₅4 = 625

Rules of Logarithms

Let us take "a" as a positive number such that "a" does not equal to 1, then let "n" be a real number and let "u and "v" be a positive real numbers.

Below you could see the logarithms rules,

1) Log_a (uv) = Log_a (u) + Log_a (v)

2) Log_a ($\frac{u}{v}$) = Log_a (u) - Log_a (v)

3) Log_a (u)ⁿ = nLog_a (u)

Laws of Logarithms

Product Law 1: The logarithm of a product is equal to the sum of the logarithms of the factor,

log_a(m X n) = log_a m + log_a n

Quotient Law 2: The logarithm of the quotient of two numbers is equal to the difference between the logarithms of the numerator and the logarithm of the denominator.

log_a`(m / n)` = log_a m - log_a n

Power Law 3: The logarithm of a power of a number is the logarithm of the number multiplied by its power exponent.

log_a `(m)^n` = n log_a m

Corollary : The logarithm of a root of a number is the logarithm of the number divided by the index of the root.

log _a `root(n)(m)` = log _a `(m^(1/n))` = `1/n` log_a m

Law 4 : Change of base rule : log _a x = log _b x . log _a b

Corollary : 1. Putting x = a in 4 log _a a = log _b a . log _ab

log _b a . log _a b = 1

log _ba = `1/(loga b )`

Common Logarithm

Logarithms calculated to the base 10 are called common logarithms. The logarithmic tables available and are computed with base 10. Therefore they are called common logarithmic tables.

Characteristic and Mantissa.

Every logarithm has two parts: The integral part are also called characteristic and the fractional parts which is expressed in decimal fraction and are also called mantissa.

Given number	Characteristic of their logarithms	Explanation
53	1	one less than the number of digit to the
2786	3	left of the decimal point.
384.6	2
7.06	0
0.7	-1	One more than the number of
0.04	-2	zeroes to the right immediately after
0.00501	-3	the decimal point

Given number	Characteristic	Mantissa	Logarithm
5978	3	0.776 6	3.776 6
597.8	2	0.776 6	2.776 6
0.005978	-3	0.776 6	-3+.776 6

Base 10 Logarithms

Here, we only use the common logarithms, because value of log e is nothing but value of log base 10 of e, tha is log₁₀ e.

Base 10 logarithms or common logarithms:

Here, base "a" is 10.

Therefore, 10^x = m,

Then x is the logarithm of m to the base 10 and it is written as, log₁₀m , this is called the common logarithms.

Normally the above common logarithms is written by log m. That is , if log₁₀m = x ⇒ then 10^x = m.

Here, we know that value of 'e' (exponents) is 2.7182818 (rounded to 7 digits). Therefore, the value of log e as follow as,

log₁₀ e = log e = log (2.7182818) = 0.434294482 .

Expanding Logarithms

We have to use the logarithmic rules to expand the logarithmic function:

The following table to show the laws for use of common logarithms.

Law	Common Logarithms
Product Rule	Log ( A x B ) = Log A + Log B
Quotient Rule	Log ($\frac{A}{B}$) = Log A - Log B
Power Rule	Log ( A )B = B( Log A )
Reciprocal Rule	Log A = $\frac{1}{Log_A 10}$

Solved Example

Question: Expand the following function log₁₀ (6). Find the value of the given logarithmic function.
Solution:

Given function is log ₁₀ (6)

We can split 6 = 3 X 2

log ₁₀ (6) = log ₁₀ 3 + log ₁₀ 2

log ₁₀ 3 = 0.4771 and log ₁₀ 2 = 0.3010

So log 10 (6) = 0.4771 + 0.3010

log ₁₀ (6). = 0.7781

Here after expanding the logarithms we will plug the values to find the value.

Adding Logarithms

Below you could see the adding logarithms examples

Solved Examples

Question 1: Add: ln 3 + ln 4
Solution:

ln 3 + ln 4 = ln 12

= 2.48490665

Question 2: Add: ln 3 + ln 4 + ln 5
Solution:

ln 3 + ln 4 + ln 5 = ln (3 x 4 x 5)

= ln (60)

Subtracting Logarithms

Below you could see subtracting logarithms examples -

Solved Examples

Question 1: Subtraction log₂ $\frac{64}{4}$

Solution:

Given log₂ $\frac{64}{4}$

= log₂(64) - log₂(4)

= 6 - 2

= 4

Question 2: Subtraction log₃ $\frac{81}{9}$
Solution:

Given log₃ $\frac{81}{9}$

= log₃ (81) - log₃(9)

= 4 - 2

= 2

Solving Logarithms

Below are the logarithm examples -

Solved Examples

Question 1: Find log 756.8
Solution:

The characteristic = 3-1 = 2

To find the mantissa, refer to logarithms table. Neglecting the decimal point, the number obtained is 7 568. Look up for the number 75 in the extreme left-hand column.

In the horizontal line of 75 and under 6(the next digit in the number), the number found is 8 785. In the same horizontal line and under 8 (the next i.e. `4^(th)` digit in the given number), the number found is 5. Adding 5 to the already obtained number 8785, we get 8790. Thus the mantissa is 0.8790.

log 756.8 = 2.8790 (8785 + 5 = 8790)

Similarly log 75.68 = 1.8790

log 0.075 68 = -2 + 0.8790

Question 2: Find log 11.648
Solution:

The Characteristic = 2-1 = 1

Leaving the decimal point, the given number consists of 5 digits. Condensing it to a four- digit number, by the rule of approximation, we get the number 1165. To get mantissa we follow the same procedure as given in examples 1. In the horizontal line of 11 and under 6, the number found is 0.0645(do not omit zero). In the horizontal line of 0645 and under 5 in M.D.columns, the number obtained is 19. Adding 19 to 0645, we get 0664. therefore the mantissa is 0.0664.

Therefore, log 11.648 = log 11.65 = 1.066 4 ( 0645 + 19 = 0664)