Home
Class 12
MATHS
If a,b,c are in G.P. and log(c)a,log(b)c...

If a,b,c are in G.P. and `log_(c)a,log_(b)c,log_(a)b` are in A.P., then the common differenec of the A.P. is

Text Solution

Verified by Experts

Given that a,b,c are in G.P.,
Then `a=b/r,b=b,c=br`, where r is common ratio
Also given that `log_(c)a,log_(b)c,log_(a)b` are in A.P.
`rArrd=log_(b)c-log_(c)a=log_(a)b-log_(b)c`
`rArrd=(log_(10)c)/(log_(10)b)-(log_(10)a)/(log_(10)c)=(log_(10)b)/(log_(10)a)-(log_(10)c)/(log_(10)b)`
`rArrd=((log_(10)c)^(2)-log_(10)alog_(10)b)/(log_(10)blog_(10)c)`
`=((log_(10)b)^(2)-log_(10)alog_(10)c)/(log_(10)alog_(10)b)`
`rArrd=((log_(10)c)^(2)-log_(10)alog_(10)b-(log_(10)b)^(2)+log_(10)alog_(10)c)/(log_(10)blog_(10)c-log_(10)alog_(10)b)`
`rArrd=((logc-logb)(logc+logb+loga))/(logb(logc-loga))`
`=(logc/bcdotlogabc)/(logc/acdotlogb)`
`=(logr)/(logr^(2))cdot(logb^(3))/(logb)`
`=3/2`
Promotional Banner

Topper's Solved these Questions

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise SOLVED EXAMPLES 5.4|1 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise SOLVED EXAMPLES 5.5|1 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise SOLVED EXAMPLES 5.2|1 Videos

  • PROBABILITY II

    CENGAGE PUBLICATION|Exercise MULTIPLE CORRECT ANSWER TYPE|6 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Archives (Numerical Value Type)|3 Videos

Similar Questions

Explore conceptually related problems

If a. b, c are in G.P. and log_ca , log_b c , log_a b are in AP show that the common difference of the AP is 1(1/2)

Three unequal positive numbers a, b, c are such that a, b, c are in G.P. while log ((5c)/(2a)), log((7b)/(5c)), log((2a)/(7b)) are in A.P. Then a, b, c are the lengths of the sides of

If a, b, c are in H.P., prove that, a(b+c), b(c+a), c(a+b) are in A.P.

if a ,b ,c are in G.P.,then (log)_a 10 ,(log)_b10,(log)_c 10 are in____

If log2,log(2^n-1) and log(2^n+3) are in A.P. then n =

If a,b,c are three unequal numbers such that a,b,c are in A.P. and (b-a) ,. (c-b) ,a are in G.P., then the value of a:b: c is-

If x ,2y ,3z are in A.P., where the distinct numbers x ,y ,z are in G.P, then the common ratio of the G.P. is a. 3 b. 1/3 c. 2 d. 1/2

a,b and c are in A.P. Prove that b+c , c+a and a+b are in A.P.

Three unequal numbers a, b, c are in A.P. and a, (b-a), (c-a) are in G.P. Prove that, a : b : c = 1 : 3 : 5 .

If a^(2), b^(2), c^(2) are in A.P., show that, (b+c), (c+a), (a+b) are in H.P.