# Logarithmic Inequalities

## 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.