To solve logarithmic inequalities, we should be aware of properties of logarithm and rules followed in inequalities.
Conversion between logarithmic form to exponential form :
Note :
The problem can be solved using change base rule, the detailed example is given below.
Solve the inequality.
Problem 1 :
Solution :
Solution is (125, ∞).
Problem 2 :
Solution :
The solution is (0, 64]
Problem 3 :
Solution :
Solution is [4, ∞).
Problem 4 :
Solution :
Solution is [4, ∞).
Problem 5 :
Solution :
Solution is [64, ∞).
Problem 6 :
Solution :
Solution is [81, ∞).
Problem 7 :
Solution :
Solution is (0, 108)
Problem 8 :
10x + 3 > 0 10x > -3 x > -3/10 |
7x - 4 > 0 7x > 4 x > 4/7 |
By considering all three, the solution for the given inequality should be (4/7, ∞).
Problem 9 :
log x + log (2 - x) < 1
Solution :
log x + log(2 - x) < 1
log [x(2 - x)] < 1
Moving base to the other side of the inequality sign, we get
x(2 -x) < 10^{1}
2x - x^{2} < 10
-x^{2} + 2x - 10 < 0
x^{2} - 2x + 10 > 0
This quadratic inequality cannot be solved using the method of factoring.
a = 1, b = -2 and c = 10
b^{2} - 4ac = (-2)^{2} - 4(1)(10)
= 4 - 40
= -36 < 0
Domain is (0, 2). So, the solution is (0, 2).
Problem 10 :
Solution :
Domain of log_{2} (x^{2} - x - 6)
x^{2} - x - 6 > 0
(x - 3) (x + 2) > 0
x > 3 and x > -2
Domain of the function is (-2, ∞) u (3, ∞), solution by solving the logarithmic inequality is (-∞, 7).
By considering all these three, we get the solution as (3, 7).
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM