Laws of exponents and logarithms

IB Mathematics: Analysis and Approaches SL· 1.6 Exponents and logarithms· 20 min read

1. Laws of Exponents★★☆☆☆⏱ 8 min

📘 Definition

Exponent Law

$a^{m}, a^{n}$

Rules governing combinations of exponents for the same non-zero base $a$ , derived from basic repeated multiplication.

Example:

$a^{m} \times a^{n} = a^{m + n}$

$a^{m} \times a^{n} = a^{m + n}$ (same base, multiply: add exponents)
$\frac{a ^{m}}{a ^{n}} = a^{m - n}$ (same base, divide: subtract exponents)
$(a^{m})^{n} = a^{mn}$ (power of a power: multiply exponents)
$(ab)^{n} = a^{n} b^{n}$ (power of a product: distribute exponent)
$(\frac{a}{b})^{n} = \frac{a ^{n}}{b ^{n}}$ (power of a quotient: distribute exponent)
$a^{0} = 1$ for $a \neq = 0$
$a^{- n} = \frac{1}{a ^{n}}$ (negative exponent = reciprocal)
$a^{\frac{m}{n}} = n a^{m} = (n a)^{m}$ (fractional exponent = nth root)

📐 Worked Example

Simplify $\frac{( 4 x ^{3} y ^{- 2} ) ^{3}}{2 x ^{2} y ^{4}}$ fully, with positive exponents only.

1
Distribute the exponent 3 across all terms in the numerator using the power of a product and power of a power rules:
2
$(4^3)(x^3)^3(y^{-2})^3 = 64x^{9}y^{-6}$
3
Rewrite the full expression and simplify term-by-term:
4
$\frac{64x^9 y^{-6}}{2x^2 y^4}$
5
Divide constants: $64/2 = 32$ . Subtract exponents for $x$ : $9 - 2 = 7$ . Subtract exponents for $y$ : $- 6 - 4 = - 10$ :
6
$32x^7 y^{-10}$
7
Rewrite with a positive exponent for the final result:
8
$\frac{32x^7}{y^{10}}$

Exam tip:

Double-check the sign of exponents after simplifying negative exponents, this is a common marking point.

2. Exponential and Logarithmic Conversion★★☆☆☆⏱ 6 min

📘 Definition

Logarithm

$lo g_{a} x$

The logarithm of $x$ with base $a$ is the exponent that $a$ must be raised to, to get $x$ . Valid for $a > 0, a \neq = 1, x > 0$ .

Example:

$lo g_{3} 9 = 2$ , since $3^{2} = 9$

Logarithms are the inverse operation of exponentiation. Any exponential statement can be rewritten as a logarithmic statement, and vice versa, using the core inverse relationship:

a^b = c \iff \log_a c = b

📐 Worked Example

(a) Write $5^{3} = 125$ in logarithmic form. (b) Write $lo g_{4} 64 = 3$ in exponential form.

1
For (a): Identify $a = 5$ , $b = 3$ , $c = 125$ . Substitute into the conversion rule $lo g_{a} c = b$ :
2
Result: $lo g_{5} 125 = 3$
3
For (b): Identify $a = 4$ , $b = 3$ , $c = 64$ . Substitute into the conversion rule $a^{b} = c$ :
4
Result: $4^{3} = 64$

3. Laws of Logarithms★★★☆☆⏱ 6 min

All logarithm laws are derived directly from the inverse relationship between logarithms and exponents, and match the corresponding laws of exponents. Below are the core laws you need for IB AA SL:

Product rule: $lo g_{a} (x y) = lo g_{a} x + lo g_{a} y$
Quotient rule: $lo g_{a} (\frac{x}{y}) = lo g_{a} x - lo g_{a} y$
Power rule: $lo g_{a} (x^{n}) = n lo g_{a} x$
Change of base rule: $lo g_{a} b = \frac{l o g _{c} b}{l o g _{c} a}$ for any $c > 0, c \neq = 1$
Special cases: $lo g_{a} 1 = 0$ , $lo g_{a} a = 1$ , $lo g_{a} (1/ x) = - lo g_{a} x$

📐 Worked Example

Simplify $3 lo g_{2} 6 - lo g_{2} 27$ into a single number.

1
Apply the power rule to the first term to move the 3 inside the logarithm:
2
$3 \log_2 6 = \log_2 (6^3) = \log_2 216$
3
Rewrite the full expression using the quotient rule for logarithms:
4
$\log_2 216 - \log_2 27 = \log_2 \left(\frac{216}{27}\right) = \log_2 8$
5
Simplify $lo g_{2} 8$ : since $2^{3} = 8$ , the final result is 3.

4. Common Pitfalls

Wrong move:

$(a^{m})^{n} = a^{m^{n}}$ (raising $m$ to the power $n$ instead of multiplying)

Why:

Confusion between the power of a power rule and standard order of operations

Correct move:

$(a^{m})^{n} = a^{m \times n}$

Wrong move:

$a^{m} + a^{n} = a^{m + n}$

Why:

The exponent addition rule only applies to products of terms with the same base, not sums

Correct move:

$a^{m} \times a^{n} = a^{m + n}$ ; $a^{m} + a^{n}$ cannot be simplified to a single power of $a$

Wrong move:

$lo g_{a} (x + y) = lo g_{a} x + lo g_{a} y$

Why:

Incorrectly distributing the logarithm over addition; the product rule only applies to products

Correct move:

$lo g_{a} (x y) = lo g_{a} x + lo g_{a} y$ ; there is no general simplification for $lo g_{a} (x + y)$

Wrong move:

$\frac{l o g _{a} x}{l o g _{a} y} = lo g_{a} (\frac{x}{y})$

Why:

Confusion between the quotient rule (for a quotient inside the log) and the change of base rule

Correct move:

$lo g_{a} (x / y) = lo g_{a} x - lo g_{a} y$ ; $\frac{l o g _{a} x}{l o g _{a} y} = lo g_{y} x$

Wrong move:

$- 2^{2} = 4$

Why:

Order of operations error: the exponent applies only to the base, not the negative sign unless enclosed in parentheses

Correct move:

$- 2^{2} = - (2^{2}) = - 4$ , and $(- 2)^{2} = 4$

5. Quick Reference Cheatsheet

Rule Type	Exponent Form	Logarithm Form
Product	$a^{m} a^{n} = a^{m + n}$	$lo g_{a} (x y) = lo g_{a} x + lo g_{a} y$
Quotient	$a^{m} / a^{n} = a^{m - n}$	$lo g_{a} (x / y) = lo g_{a} x - lo g_{a} y$
Power	$(a^{m})^{n} = a^{mn}$	$lo g_{a} x^{n} = n lo g_{a} x$
Zero Rule	$a^{0} = 1 (a \neq = 0)$	$lo g_{a} 1 = 0$
Identity	$a^{1} = a$	$lo g_{a} a = 1$
Negative	$a^{- n} = 1/ a^{n}$	$lo g_{a} (1/ x) = - lo g_{a} x$
Special	$a^{m / n} = n a^{m}$	$lo g_{a} b = \frac{l o g b}{l o g a}$ (Change of Base)

6. Frequently Asked

Do I need to memorize all of these laws?

Yes, these laws are fundamental to almost all other topics in IB AA SL, so regular practice will help you memorize them.

Can I use a calculator for these problems?

You may use a calculator for Paper 2, but Paper 1 requires you to simplify expressions without a calculator, so you must memorize the rules.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Simplify exponential expression
2023 · 2
Solve logarithmic equation
2021 · 1
Combine logarithmic terms

Going deeper

syllabusIB AA SL Official Topic 1.6 ReferenceFull syllabus overview

What's Next

Mastering the laws of exponents and logarithms is an essential foundational skill for almost all subsequent topics in IB AA SL. You will use these rules constantly when solving equations, graphing functions, manipulating sequences and series, and even working with calculus topics like differentiation and integration of exponential functions later in the course. Even topics like trigonometry and complex numbers rely on correct exponent manipulation, so it is critical to be fluent with these rules before moving on. Next, you will apply these laws to solve exponential and logarithmic equations, before exploring the properties of exponential and logarithmic functions themselves.