IB Mathematics AI SL Laws of exponents Study Notes - New Syllabus

EXPONENTS

♦ The Exponential \( 2^x \)
Define \( 2^x \) for \( x \) in
1. N: \( 2^0 = 1 \), \( 2^n = 2 \times 2 \times \cdots \times 2 \) (n times)
2. Z: \( 2^{-n} = \frac{1}{2^n} \)
3. Q: \( 2^{m/n} = \sqrt[n]{2^m} \)
4. R: For irrational \( x \), \( 2^x \) is calculated using a calculator.

♦ General Definition for \( a^x \):
 \( a^0 = 1 \)
 \( a^n = a \times a \times \cdots \times a \) (n times)
 \( a^{-n} = \frac{1}{a^n} \)
 \( a^{m/n} = \sqrt[n]{a^m} \)
 For irrational \( x \), \( a^x \) is calculated using a calculator.

Notice:
 \( a^x \) is defined for \( a < 0 \) only if \( x \) is an integer.
 \( 0^x = 0 \) for \( x \neq 0 \). \( 0^0 \) is undefined.

♦ Properties of Exponents:

1. \( a^x a^y = a^{x+y} \)
2. \( \frac{a^x}{a^y} = a^{x-y} \)
3. \( (ab)^x = a^x b^x \)
4. \( \left( \frac{a}{b} \right)^x = \frac{a^x}{b^x} \)
5. \( (a^x)^y = a^{xy} \)

♦ Simple Exponential Equations
If \( a \neq 1 \), then:
$ a^x = a^y \implies x = y $