Basic Mathematics and Logarithm

Logarithm

Definition
$lo g_{a} x = y ⟺ a^{y} = x,$ $a > 0, a \neq = 1, x > 0$
Product and quotient
$lo g_{a} (mn) = lo g_{a} m + lo g_{a} n,$ $lo g_{a} \frac{m}{n} = lo g_{a} m - lo g_{a} n$
Power
$lo g_{a} m^{k} = k lo g_{a} m,$ $lo g_{a^{k}} m = \frac{1}{k} lo g_{a} m$
Change of base
$lo g_{a} m = \frac{lo g _{b} m}{lo g _{b} a},$ $lo g_{a} b \cdot lo g_{b} a = 1$
Inverse
$a^{l o g_{a} x} = x,$ $a^{l o g_{b} c} = c^{l o g_{b} a}$
Inequalities
$a > 1 : lo g_{a} x > lo g_{a} y ⟺ x > y > 0;$ $0 < a < 1 : lo g_{a} x > lo g_{a} y ⟺ 0 < x < y$

Modulus
$∣ x ∣ = {x, - x, x \geq 0 x < 0,$ $x^{2} = ∣ x ∣$
Modulus inequalities
$∣ x ∣ < a ⟺ - a < x < a,$ $∣ x ∣ > a ⟺ x < - a$ or $x > a (a > 0)$
Triangle inequality
$∣ a ∣ - ∣ b ∣ \leq ∣ a \pm b ∣ \leq ∣ a ∣ + ∣ b ∣$
Wavy curve method
$\frac{( x - a _{1} ) ( x - a _{2} ) \dots}{( x - b _{1} ) ( x - b _{2} ) \dots} > 0$
Mark the roots on a number line; the sign alternates between consecutive roots (a root of even power does not change the sign). Exclude the $b_{i}$ .