If log 2 , log (x power 2 - 1) and log (x power 2 + 3 ) are in A.P. Then find x?

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Sujoy Das

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square root 5 and square root( -1)
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Given:- log2, log and log are in AP. To find: Values of x satisfying the above relation. Solution:- Step 1 If a,b,c are in AP then b - a = c - b. Step 2 for ease of solving let t=x^2 So, log(t-1) - log2 = log(t+3) - log(t-1) ..........(1) Step 3 log a - log b = log(a/b) applying...
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Given:- log2, log[(x^2)-1] and log[(x^2)+3] are in AP. To find: Values of x satisfying the above relation. Solution:- Step 1 If a,b,c are in AP then b - a = c - b. Step 2 for ease of solving let t=x^2 So, log(t-1) - log2 = log(t+3) - log(t-1) ..........(1) Step 3 log a - log b = log(a/b) applying this property to equation (1) log [(t -1)/2] = log [(t+3)/(t-1)]. Step 4 Taking anti-log on both sides, (t-1)/2 = (t+3)/(t-1), Step 5 Cross multiplying (t^2) - 2t + 1 = 2t + 6 taking all terms towards LHS t^2 - 4t -5 = 0 Step 6 Finding roots of this equation t^2 + (-5+1)t + [(-5)*1] = 0 (t-5)*(t+1) = 0 So, t = 5 and t = -1. Step 7 As we assumed before t = x^2 Case 1 t = 5 so x^2 = 5 therefore, x = +sqrt(5) or -sqrt(5) Case 2 t=-1 so x^2 = -1 therefore, x = +i or -i. So final values of x satisfying the above relation are x = +sqrt(5), -sqrt(5), +i, -i. NOTE:- Here sqrt() denotes the square root operation. 'i' denotes sqrt(-1), which is an imaginary number. read less
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Mathematics for JEE Mains/Advanced, XI & XII (All Boards)

x² = 5
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GMAT Math Expert

square root 5 and i (square root( -1))
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Math Educator for Std.11th ,12th , Engineering Entrance and Degree Level with 11+ Years Experience

If a,b,c are in A.P. Then b = a+c/2 , since log 2 , log (x power 2 - 1) and log (x power 2 + 3 ) are in A.P. then log 2 + log (x power 2 + 3 ) / 2 = log (x power 2 - 1) ,after solving , x power 2 = 5 hence, x
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Math Educator for Std.11th ,12th , Engineering Entrance and Degree Level with 11+ Years Experience

if a,b,c are in A.P. then b = a+c/2 . since log 2 , log (x power 2 - 1) and log (x power 2 + 3 ) are in A.P. then log 2 + log (x power 2 + 3 ) / 2 = log (x power 2 - 1) OR 2 (x power 2 + 3 ) = 2 power (x power 2 - 1) after solving , x power 2 = 5 ,Hence x = log 5
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square root 5 and square root( -1)
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Specialization in Organic Chemistry. Teaching Chemistry has always been my passion.

X may be +/- Sq. Root of 5 or i, where i is iota.
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X =?5
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x=1
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