Find the solution set in log2(x−1)+log2(x−2)>1
x>3
log2(x−1)+log2(x−2)>1
For Logarithm to be defined
x - 1 > 0, x > 1 ------------------(1)
And x - 2 > 0, x > 2 ------------(2)
This inequality can also be written as
log2(x−1)(x−2) > 1
log2(x2−3x+2) > 1
Since, base of the logarithm greater than 1. So, given logarithm is a decreasing function.
x2−3x+2>21
x2−3x+2−2>0
x2−3x>0
(x)(x−3)>0 x∈(−∞,0)∪(3,∞) ---(3)
So,
From equation 1,2 and 3 Or common part of equation 1,2 and 3
We get,
x > 3