Logarithm Formulas Sheet | Properties of Logarithms | Log Rules | Lows of Log

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 MantissaRule for write Mantissa and CharacteristicNatural 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 = 10n and 254 = 10m

Now rewrite above exponential forms into logarithmic forms

i.e n = log 10 524  and m = log 10 254

524 x 254 = 10n x 10m

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 (n1n2n3n4n5 – – – – – – –  nk) = log n1 + log n2 + log n3 + log n4 + log n5   +  – – – – – –  + log nk

(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 = 10m and 254 = 10n

Now rewrite above exponential forms into logarithmic forms

i.e m = log 10 524  and n = log 10 254

524 /254 = 10m x 10n

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   ;

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

For example –   Take 528

Let 52 = 10n    ——  ( 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 101

Now  528 = (5.2 x 101 )8 = 5.28 x 108

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

528 = (10n )8

⇒ 528 = 108n

Now rewrite above exponential forms into logarithmic forms

8 n   = log 10 (528)

Substituent the value of ‘n’ in the above equation

8 log 10 52  = log 10 (528)

Base Change Rules

1.  log _{n}m = \frac{1}{log _{m}n}

2.  log n m = log a m / log a n   = log a m .  log n a

3.  log _{(n^b)}m = \frac{1}{b} \ \ log _{n}(m)

Some other Properties of logarithms

3. a log a N = N

4.  log a a = 1

5. log _{n}m = \frac{log _{10}m}{log _{10}n} = \frac{log \ m}{log \ n}

6. log _{(n^b)}(n^a) = \frac{a}{b}

7. log _{(n^b)}(m^a) = a \ log _{(n^b)}(m) = \frac{a}{b} \ \ log _{n}(m)

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

  Logarithm formulas sheet| log rules |properties of logarithms | logarithm rules practice | logarithm tutorial

 

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My self Sivaramakrishna Alluri. Thank you for watching my blog friend

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