If log 18 to the base 12=a and log 54 to the base 24=b,prove that ab+5(a-b)=1?

2 Answers
Jul 11, 2017

see explanation

Explanation:

Please check to make sure I've read the problem correctly. Logarithms may be written wrong...

  1. log12(18)=a
  2. log24(54)=b

ab+5(ab)?=1
(log12(18)log24(54))+5(log12(18)log24(54))=1

Jul 12, 2017

Please see below.

Explanation:

As log12(18)=a, we have 12a=18

i.e. (22×3)a=2×32

or 22a1×3a2=1

similarly as log24(54)=b, we have 24b=54

i.e. (23×3)b=2×33

or 23b1×3b3=1

Comparing the two 2a1=3b1 or 2a3b=0 ........(A)

and a2=b3 or ab+1=0 i.e. 2a2b+2=0 ........(B)

Subtracting (A) from (B), we get

b+2=0 i.e. b=2

and a=b1=3

hence ab+5(ab)

=(3)×(2)+5×(1)=65=1

(Note that (B) gives ab=1.)