Paper 4: Fundamentals of Business Mathematics & Statistic

2.24 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Algebra

CHANGE OF BASE :

The relation between the logarithm of a number of different bases is given by

Loga m = logb m × loga b.

Let x = loga m, y = logb m, z = loga b, then from definition ax = m, by = m, az = b.

Hence ax = m = by

Loga m = logb m x loga b.

Cor.l. loga b x logb a = 1. This result can be obtained by putting m = a in the previous result, loga a =1.

Cor. 2. loga m = logb m/logb a.

Let x = loga m, ax = m ; take log to the base b we find x logba = logb m.

∴ x = logbm / logb a.

Hence the result.

SOLVED EXAMPLES :

Example 32 : Find the logarithm of 2025 to the base3 5..

Solution :

Let x be the required number ; then (3 5)x = 2025 = 3^4 .5^2 = (3 5)^4 ∴ x = 4.

∴ 4 is the required number.

Example 33 : The logarithm of a number to the base 2 is k. What is its logarithm to the base 2 2?

Solution :

Let ( 2)x = N.

Since 2 2 = 2.21/2 = 23/2

So 2 (2 ) (2 2)= 3/2 1/3= 1/3

∴(2 2)k/3^ = N.

∴ the reqd. number is k 3.

Example 34 : Find the value of log 2 [log 2 {log 3 (log 3 273 )}].

Solution :

Given expression

= log 2 [log 2 {log 3 (log 3 39 )}] = log 2 [log 2 {log 3 (91og 3 3)} ]

= log 2 [log 2 {log 3 9}] (as log 3 3 = l)

= log 2 [log 2 {log 3 32 }] = log 2 [log 2 {2log 3 3}] = log 2 [log 2 2] = log 2 1 = 0

Example 35 : If log 2 x + log 4 x + log 16 x =^214 , find x

Solution :

log 2 16 x log 16 x + log 4 16 x log 16 x + log 16 x =^214

or 41og 16 x + 21og 16 x + log 16 x=^214 or 71og 16 x=^214