In this session explained about **logarithm formulas** ( Laws of Logarithms) like logarithm addition rule and subtract rule and *Base change rules* and results on *logarithmic inequalities* with examples

## Log Formula | Logarithm Rules Practice | Logarithm Tutorial | Exercise – 2

Please go through the below link for **basic concepts of logarithms** like *Definition of Logarithm* , *Characteristic and Mantissa* , *Rule for write Mantissa and Characteristic* , *Natural Logarithm , Common Logarithms* .. etc

**Logarithm Tutorial | Exercise – 1**

### Logarithm formula sheet ( Laws of Logarithms )

#### First Law of the logarithms – ( logarithm addition rule )

**” Log of product = Sum of Logs”
**

**log _{a }m n = log _{a }m + log _{a }n** ;

Where **m, n** & **a** are positive real numbers and a ≠ 1

**For example** – Take 524 x 254

Let 524 = 10^{n} and 254 = 10^{m}

Now rewrite above exponential forms into logarithmic forms

i.e n = log _{10} 524 and m = log _{10} 254

**⇒** 524 x 254 = 10^{n }x 10^{m}

**⇒ **524 x 254 = 10 ^{n+m}

Now rewrite above exponential forms into logarithmic forms

n + m = log _{10} (524 x 254)

Substitute ‘**n**‘ and **‘m**‘ vales in the above equation

log _{10} 524 + log _{10} 254 = log _{10} (524 x 254)

**Generalization of the above logarithmic law**

log (abc) = log a + log b + log c

log (n_{1}n_{2}n_{3}n_{4}n_{5} – – – – – – – n_{k}) = log n_{1} + log n_{2} + log n_{3} + log n_{4} + log n_{5 } + – – – – – – + log n_{k}

(Note : As per log rules, If base is not mentioned for any logarithmic equations then it is considered as base **10.** The logarithmic calculation to base 10 are called common logarithms.)

#### Second Law of the logarithms – ( logarithm subtract rule )

” log of quotient = difference of logs”

**log _{a }(m/n) = log _{a }m – log _{a }n ;**

Where **m, n **& **a** are positive real numbers and a ≠ 1

**For example **– Take 524 / 254

Let 524 = 10^{m} and 254 = 10^{n}

Now rewrite above exponential forms into logarithmic forms

i.e m = log _{10} 524 and n = log _{10} 254

**⇒** 524 /254 = 10^{m }x 10^{n}

**⇒ **524 / 254 = 10 ^{m -n }

Now rewrite above exponential forms into logarithmic forms

m – n = log _{10} (524 /254)

Substitute **‘n‘** and ‘**m**‘ vales in the above equation

log _{10} 524 – log _{10} 254 = log _{10} (524/254)

#### Third Law of the logarithms –

**log _{a }(m ^{n}) = n log _{a }m ** ;

Where m, n and a are positive real numbers and a ≠ 1

**For example **– Take 52^{8}

Let 52 = 10^{n } —— ( 1)

Now rewrite above exponential forms into logarithmic forms

i.e n = log _{10} 52 —– ( 2)

Here 52 can be written as 5.2 x 10^{1}

Now 52^{8} = (5.2 x 10^{1} )^{8} = 5.2^{8} x 10^{8}

Raising the power to** 8** on both sides of the equation ( 1)

52^{8} = (10^{n })^{8}

**⇒ 52 ^{8} = 10^{8n }**

Now rewrite above exponential forms into logarithmic forms

8 n = log _{10} (52** ^{8}**)

Substituent the value of ‘**n’** in the above equation

8 log _{10} 52 = log _{10} (52** ^{8}**)

#### Base Change Rules

1.

2. **log _{n} m = log _{a} m / log _{a} n = log _{a} m . log _{n} a**

3.

##### Some other Properties of logarithms

3. **a log _{a} N = N**

4. **log _{a} a = 1**

5.

6.

7.

**Logarithmic Inequalities**

1. If n > 1 and log _{n} a > log _{n} b then a > b

2. If n <1 1 and log _{n} a > log _{n} b then a < b

Please go through the below link for **logarithm applications with examples and solutions**

**Logarithm Tutorial | Exercise – 3**

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