Theory of Logarithmics

Introduction

Consider the statement 2³ = 8. In English words this is read as the base 2 raisen to the power 3 gives 8.

By using a different emphasis, the same information can be given. The power to which the base 2 must be raised to give 8 is 3. This is written as log₂8 = 3, in Mathematical format

Examples :

	103 = 1000 Þ log101000 = 3	[Note : Log10 is also written as log {without writting and base} or lg].
	e3 = 20.086... Þ loge20.086... = 3	[Note : Loge is also written as ln].
	xy = z Þ logxz = y	[Note : the base of a log can be any number greater than zero (0), i.e. x > 0].
[Note : Only log10 and loge have a different way of presentation. All other bases have to be stated].

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Laws of Logarithms

Law for log_a(bc)

Consider x = log_ab and y = log_ac
              Þ  a^x = b and Þ   a^y= c

Now bc = a^x.a^y
         bc = a^x+y

     Þ   log_a(bc) = x + y
     Þ   log_a(bc) = log_ab + log_ac

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Law for log_a()

Consider x = log_ab and y = log_ac
              Þ  a^x = b and Þ   a^y= c

     
         = a^{x - y}

     Þ   log_a = x - y
     Þ   log_a = log_ab - log_ac

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Law for log_abⁿ

Consider x = log_abⁿ     
              Þ  a^x= bⁿ     [By Law for log_a(bc)]
              Þ  
              Þ   = b
              Þ   = b
              Þ  log_ab =
              Þ  x = nlog_ab     [By Law for log_a()]

Þ  log_abⁿ = nlog_ab

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Changing the base of a logarithm

Changing the base from c to b

log_ca = x
     c^x = a

Taking logs to base b on both sides

    log_bc^x = log_ba
º xlog_bc = log_ba
Þ  x =

 

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