Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 2.25

or log 16 x =^34 or x = 163/4 = 24x3/4 = 2^3 = 8.

Example 36 : If p = log 10 20 and q = log 10 25, find x and such that 2 log 10 (x + 1) = 2p – q
Solution :
2p – q = 2 log 10 20 – log 10 25 = log 10 (20)^2 – log 10 25

= log 10 400 – log 10 25 =log 1040025 =log 16 10

Now, 2log (x +1)= log 10 10 16 or, log 10 (x + 1)^2 = log 10 16 or, (x + 1)^2 = 16 = (±4)^2
or, x + 1 = ± 4
∴ x = 3, – 5.
Example 37 : If x = log2a a, y = log3a 2a, z = log4a 3a, Show that : xyz + 1 = 2yz.
Solution :
L.H.S. = log2a a. log3a 2a. log4a 3a + 1
=(log a log 10. log 2a log 10. log 3a log 10 1 10 × 2a )( 10 × 3a )( 10 × 4a )+

10 10 10

10 10 10

log a log 2a log 3a 1

=log 2a log 3a log 4a× × +

(^10) 4a 4a 4a 4a 2
10
log a 1 log a log 4a log (a 4a) log 4a
log 4a+ = + = ⋅ =
R.H.S. = 2log3a 2a. log4a 3a = log4a (2a)^2 = log4a. 4a^2
Hence the result.
Example 38 : Show that log 3 3 3..... 1. 3 ∞ =
Solution :
Let, x 3 3 3 ...= or x 3 3 3...^2 = ∞
(squaring both sides)
or, x^2 = 3x or, x^2 – 3x = 0 or, x (x – 3) = 0 or, x – 3 = 0 (as x ≠ 0),
∴ x = 3
∴ given expression = log 3 x = log 3 3 = 1.
SELF EXAMINATION QUESTIONS

1. If^12 log 3 M + 3loga N = 1, express M in terms of N. [Ans. 9N–6]


2. If a^2 + b^2 = 7ab, show that :
   (i) 2 log (a – b) = log 5 + log a + log b.