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 = 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 the 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 the 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 the above exponential forms into logarithmic forms
m – n = log 10 (524 /254)
Substitute ‘n‘ and ‘m‘ values 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 528
Let 52 = 10n —— ( 1)
Now rewrite the 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 the 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.
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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