Transforming the base of logarithms, Mathematics

Assignment Help:

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)


Related Discussions:- Transforming the base of logarithms

How many square centimeters are in one square meter, How many square centim...

How many square centimeters are in one square meter? There are 100 cm in a meter. A square meter is 100 cm through 100 cm. The area of this is 10,000 sq cm (100 × 100 = 10,000)

What is equivalence relation, What is equivalence relation?  Prove that rel...

What is equivalence relation?  Prove that relation  'congruence modulo' (  ≡mod m) is an equivalence relation.  Ans: A relation R illustrated on a nonempty set A is said to be

Quantitative Techniques, The following table given the these scores and sal...

The following table given the these scores and sales be nine salesman during last one year in a certain firm: text scores sales (in 000''rupees) 14 31 19

Callie grandmother pledged $0.50 for each mile callie walked, Callie's gran...

Callie's grandmother pledged $0.50 for each mile Callie walked in her walk-a-thon. Callie walked 9 miles. How much does her grandmother owe? Multiply the number of miles (9) th

ADDING AND SUBTRACTING EQUATION, GUESS THE NUMBER THAT WHEN YOU SUBTRACT 6 ...

GUESS THE NUMBER THAT WHEN YOU SUBTRACT 6 AND THEN SUBTRACT 0 IS-14

Solving whole-number riddles, I am greater than 30 and less than 40. The su...

I am greater than 30 and less than 40. The sum of my digits is less than 5. who am I?

Arthemetic progreession, ball are arranged in rows to form an equilateral t...

ball are arranged in rows to form an equilateral triangle .the firs row consists of one abll,the second of two balls,and so on.If 669 more balls are added,then all the balls canbe

Mashed patatos, I have 6 cups of patatos that I have to share with 13 frien...

I have 6 cups of patatos that I have to share with 13 friends write that as the nearest hundredth

Reflection matrix, how do i solve reflection matrix just looking at the num...

how do i solve reflection matrix just looking at the numbers in a matrix

Calculus, I need help with my calculus work

I need help with my calculus work

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd