Theory

Logarithmic equations and inequalities

The logarithm as an exponent, its properties and its function, and the methods for solving logarithmic equations and inequalities starting from the domain conditions.

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The logarithm
The properties of logarithms

The logarithm

Logarithm

For $a > 0$ , $a \neq = 1$ and $b > 0$ , the logarithm to base $a$ of $b$ is the exponent to which $a$ must be raised to obtain $b$ :

lo g_{a} b = c ⟺ a^{c} = b .

The logarithm exists only for positive arguments, because $a^{c} > 0$ for every $c$ . From the definition follow the two identities

lo g_{a} (a^{x}) = x (x \in R), a^{l o g_{a} x} = x (x > 0),

which are used to pass from exponential form to logarithmic form and vice versa; equations of the types $lo g_{a} A = k$ and $a^{A} = k$ are based on these identities.

Example

$lo g_{2} 8 = 3$ because $2^{3} = 8$ ; $lo g_{3} \frac{1}{9} = - 2$ because $3^{- 2} = \frac{1}{9}$ ; $lo g_{5} 1 = 0$ because $5^{0} = 1$ .

The properties of logarithms

1. Properties of logarithms

For $a > 0$ , $a \neq = 1$ and $x > 0$ , $y > 0$ :

lo g_{a} (x y) = lo g_{a} x + lo g_{a} y, lo g_{a} \frac{x}{y} = lo g_{a} x - lo g_{a} y, lo g_{a} (x^{p}) = p lo g_{a} x (p \in R) .

The three formulas hold when the arguments $x$ and $y$ are positive, therefore before applying them to expressions containing the unknown the domain conditions must be written. In equations and inequalities they are used to combine several logarithms into a single logarithm.

One example for each property

lo g_{6} 2 + lo g_{6} 3 = lo g_{6} 6 = 1, lo g_{2} 40 - lo g_{2} 5 = lo g_{2} 8 = 3, lo g_{3} (9^{5}) = 5 lo g_{3} 9 = 10.

In each case the arguments are positive numbers, so the properties apply.

2. Change of base

For $a, b > 0$ both different from $1$ and $x > 0$ :

lo g_{a} x = \frac{lo g _{b} x}{lo g _{b} a} .

It makes it possible to express any logarithm in a chosen base, for example in base $e$ or in base $10$ , which are available on a calculator. For example $lo g_{4} 8 = \frac{lo g _{2} 8}{lo g _{2} 4} = \frac{3}{2}$ .

The theory continues with 5 more sections (the logarithmic function, domain conditions, logarithmic equations…), plus all worked examples, the solutions and the PDF.

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