Theory

Exponential equations and inequalities

The function a^x and its properties, and the methods for solving equations and inequalities in which the unknown appears in the exponent.

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The exponential function
The rules of powers

The exponential function

Exponential function

For $a > 0$ , $a \neq = 1$ , the exponential function with base $a$ assigns to every $x \in R$ the number $a^{x}$ . Properties:

$a^{x} > 0$ for every $x \in R$ : the graph lies above the $x$ -axis and the line $y = 0$ is a horizontal asymptote;
$a^{0} = 1$ , so the graph passes through the point $(0, 1)$ whatever the base;
the function is injective: different exponents give different powers;
for $a > 1$ the function is strictly increasing; for $0 < a < 1$ it is strictly decreasing.

All the solution methods in this chapter rest on injectivity and monotonicity: injectivity justifies passing from the powers to the exponents in equations, and monotonicity governs the direction of the inequality sign in inequalities.

The two basic graphs

For $a = 2$ the function increases; for $a = \frac{1}{2}$ it decreases, since $(\frac{1}{2})^{x} = 2^{- x}$ .

y = 2^{x}

y = 2^{x}

: base greater than

1

, increasing function

y = \left(\dfrac{1}{2}\right)^{x}

y = (\frac{1}{2})^{x}

: base between

0

and

1

, decreasing function

The rules of powers

Rules of powers

For $a > 0$ and $x, y \in R$ :

a^{x} a^{y} = a^{x + y}, \frac{a ^{x}}{a ^{y}} = a^{x - y}, (a^{x})^{y} = a^{x y}, a^{- x} = \frac{1}{a ^{x}} .

They are used to rewrite the two sides of an equation or an inequality as powers of the same base. For example $8^{x} = (2^{3})^{x} = 2^{3 x}$ and $\frac{1}{9} = 3^{- 2}$ .

The theory continues with 5 more sections (equations with the same base, equations solved with the logarithm, equations with substitution…), plus all worked examples, the solutions and the PDF.

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