Transforming the base of logarithms, Mathematics

Suppose that we know the logarithms of all numbers which are expressed to base 'a' and we are required to find the logarithms of all these numbers to base 'b'. We proceed as follows. Let N be any one of the numbers of which we are required to find the logarithm to base 'b' and the value itself be some 'x'. That is, logbN = x or N = bx. But we already know the value of loga N. Also logaN can be expressed as loga(bx) as N = bx. By rule 5, loga(bx) can be expressed as x.logab or 

         xlogab = logaN

 

1728_transforming base of logarithm.png                   ...............(1)

Since the values of N and b are known, the values of loga N and loga b can be found from the tables. These values when substituted in equation (1) gives us the value of logb N.

In the above equation, what will happen if N = a. Equation (1) will be

1638_transforming base of logarithm1.png

 (because logaa = 1)

or

         logba x loga b = 1

Example 

Find the values of the following.

  1. log3 81

We know that 81 = 34. Therefore, log3 34 = 4. log3 = 4.1 = 4

  1. log3 (9 x 27 x 81)

         log3 (9 x 27 x 81) = log39 + log327 +  log381

                                           (loga M.N = loga M + logaN)

                                        = log3(32) + log3(33) + log3(34)

                                        = 2.log33 + 3.log33 + 4.log33

                                        = 2 + 3 + 4 = 9

  1. log3 4√35

In this example we apply an extension of rule 5.

(5/4).log33 = (5/4).1 = (5/4)

  1. log3 (243/81)

         log3243 - log381 [loga (M/N) = logaM - loga N]

         log335 - log334

         5.log33 - 4.log33 = 5 - 4 = 1

         (logM (MP) = p.logM M = p)

Posted Date: 9/13/2012 5:44:25 AM | Location : United States







Related Discussions:- Transforming the base of logarithms, Assignment Help, Ask Question on Transforming the base of logarithms, Get Answer, Expert's Help, Transforming the base of logarithms Discussions

Write discussion on Transforming the base of logarithms
Your posts are moderated
Related Questions
Five cards - the ten, jack, queen, king and ace, are well shuffled with their face downwards. One card is then picked up at random. (i)  What is the probability that the card is

what is the result if 816.537 is rounded to the nearest tenth

For a population with a mean of μ=70 and a standard deviation of o=20, how much error, on average, would you expect between the sample mean (M) and the population mean for each of

find the no of solution of 2*3*4*5*6*6

what is classification and how can you teach it?


I need help in assignment of stats? Please give me assist in my stats exam.

Assume that the amount of air in a balloon after t hours is specified by                                             V (t ) = t 3 - 6t 2 + 35 Calculate the instantaneous

Example of Least Common Denominator: Example: Add 1/7 +2 /3 + 11/12 + 4/6 Solution: Step 1:             Find out primes of each denominator. 7 = 7 (already is

HOW WE CAN FACTORISE 12X+7X+1