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

IB Mathematics AI SL Laws of exponents Study Notes

LEARNING OBJECTIVE

Laws of exponents with integer exponents.

Key Concepts:

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} $

Example

Solve the following,

1. $ 5^{-2} = \, ? $
2. $ \left( \frac{1}{5} \right)^{-2} = \, ? $
3. $ \left( \frac{3}{5} \right)^{-2} = \, ? $
4. $ 8^{2/3} = \, ? $
5. $ 27^{-4/3} = \, ? $

▶️Answer/Explanation

Solution:

1. $ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} $
2. $ \left( \frac{1}{5} \right)^{-2} = 5^2 = 25 $
3. $ \left( \frac{3}{5} \right)^{-2} = \left( \frac{5}{3} \right)^2 = \frac{25}{9} $
4. $ 8^{2/3} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4 $
5. $ 27^{-4/3} = \frac{1}{81} $

Example

1. $ a^3 a^2 $.
2. $ \frac{a^6}{a^2} $.
3. $ \frac{x^3 x^5}{x^4} $.
4. $ 2^{x+1} \cdot 2^{3x} $.
5. Express $ 8x^3 $ as a perfect cube.
6. $ \frac{x^3 y^3}{z^3} $.
7. Express $ \frac{16a^2}{b^4} $ as a perfect square.

▶️Answer/Explanation

Solution:

1. $ a^3 a^2 = a^{3+2} = a^5 $
2. $ \frac{a^6}{a^2} = a^{6-2} = a^4 $
3. $ \frac{x^3 x^5}{x^4} = \frac{x^{3+5}}{x^4} = \frac{x^8}{x^4} = x^{8-4} = x^4 $
4. $ 2^{x+1} \cdot 2^{3x} = 2^{(x+1) + 3x} = 2^{4x+1} $
5. $ 8x^3 = (2x)^3 $
6. $ \frac{x^3 y^3}{z^3} = \left( \frac{xy}{z} \right)^3 $
7. $ \frac{16a^2}{b^4} = \left( \frac{4a}{b^2} \right)^2 $

♦ Simple Exponential Equations

If $ a \neq 1 $, then:
$ a^x = a^y \implies x = y $

Example

Find the value of x

(a) $ 2^{3x-1} = 2^{x+2} $

Find the value of x

(b) $ 2^{3x-1} = 4^{x+2} $

▶️Answer/Explanation

Solution:

(a)$ 3x – 1 = x + 2 \implies x = \frac{3}{2} $

(b)$ 4 = 2^2 \implies 3x – 1 = 2x + 4 \implies x = 5 $

♦ Understanding Logarithms

Logarithms are the inverse of exponentiation. If you’ve seen equations like $ 2^x = 1096 $ or $ 3^x = \frac{1}{27} $, logarithms help us solve them. The rule is:

$ a^x = b \iff x = \log_a b $

This tells us that a logarithm answers the question: “To what power must we raise $ a $ to get $ b $?”

Conditions:

$ a > 0, \, a \ne 1, \, b > 0 $

♦ $\log_{10} x$: Common logarithm (base 10)
♦ $\log_e x = \ln x$: Natural logarithm (base e)

Example

Solve

$5^x = 125$

Show all the proper steps.

▶️Answer/Explanation

Solution:

Take logarithm base 5 on both sides:

$x = \log_5 125$

Since $5^3 = 125$, the solution is:

$x = 3$

Example

With the help of GDC

Solve

$4^x = 50$

▶️Answer/Explanation

Solution:

Take logarithm base 4 on both sides:

$x = \log_4 50$

Use the change of base rule and calculator:

$x = \frac{\log_{10} 50}{\log_{10} 4} \approx \frac{1.6989}{0.6021} \approx 2.82$

$x \approx 2.82$

♦ Natural Logarithms (Base e)

A natural logarithm, written as ln or $\log_e$, is a logarithm with base e, where $e \approx 2.71828$. These are commonly used in scientific and calculus-based applications because of their unique behavior.

Remember: $\log_e x = \ln x$

♦ Estimating Logarithms Numerically

Logarithmic values are often irrational, so calculators or computers (like a GDC) are needed to estimate their values accurately.

Example

Use calculator to find:

$\log_{10} 6 \approx ?$

$\ln 6 \approx ?$

▶️ Answer/Explanation

Common Logarithm:

$10^{0.7782} \approx 6$
$\log_{10} 6 \approx 0.7782$

Natural Logarithm:

$e^{1.7918} \approx 6$
$\ln 6 \approx 1.7918$