Logarithmic Inequalities
Standard Logarithmic Inequalities:
1. If \({{\log }_{a}}x>{{\log }_{a}}y\Rightarrow \left\{ \begin{align} & x>y,\ if\ a>1 \\ & 0<x<y,\ if\ 0<a<1 \\ \end{align} \right\}\).
2. If \({{\log }_{a}}x>y\Rightarrow \left\{ \begin{align} & x>{{a}^{y}},\ if\ a>1 \\ & 0<x<{{a}^{y}},\ if\ 0<a<1 \\ \end{align} \right\}\).
3. If logₐ x > 0 ⇒ x > 1 and a > 1 (or) 0 < x < 1 and 0 < a < 1
Frequently used Inequalities:
- (x – a) (x – b) < 0 (a < b) ⇒ a < x < b
- (x – a) (x – b) > 0 (a < b) ⇒ x < a (or) x > b
- |x| < a ⇒ – a < x < a
- |X|> a ⇒ x < -a (or) x > a
Examples 1: Solve log₂ (x – 1) > 4.
Solution: Given that,
log₂ (x – 1) > 4
x – 1 > 2⁴
x – 1 > 16
x > 16 + 1
x > 17.
Example 2: Solve log₃ (x – 2) ≤ 2.
Solution: Given that,
log₃ (x – 2) ≤ 2.
(x – 2) ≤ 3².
(x – 2) ≤ 9.
x ≤ 9 + 2.
x ≤ 11.
Example 3: Solve log₀.₃ (x² – x + 1) > 0.
Solution: Given that,
log₀.₃ (x² – x + 1) > 0.
0 < (x² – x + 1) < 0.3°.
0 < (x² – x + 1) < 1.
x² – x + 1 > 0 and
x² – x < 0
x (x – 1) < 0
0 < x < 1
As x² – x + 1 = (x – ½)² + ¾ > 0, for all real x.