# Laws of exponents and logarithms

> IB Mathematics: Analysis and Approaches SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u1-laws-of-exponents-and-logarithms/

This sub-topic covers core rules for manipulating exponential and logarithmic expressions, forming the foundation for solving equations and working with exponential functions across IB Mathematics AA SL.

**Prerequisites:** [Basic algebraic simplification and index notation](https://www.owlsprep.com/study/ib-math-aa-sl-u1-basic-algebraic-manipulation/)

## Learning objectives

- Recall and apply all exponent laws to simplify algebraic expressions
- Convert between exponential and logarithmic form correctly
- Apply core logarithm laws to combine and simplify logarithmic expressions
- Recognize and avoid common manipulation errors with exponents and logs

## Laws of Exponents

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

*Notation:* $a^m, a^n$

*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)
- $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ (power of a quotient: distribute exponent)
- $a^0 = 1$ for $a \neq 0$
- $a^{-n} = \frac{1}{a^n}$ (negative exponent = reciprocal)
- $a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$ (fractional exponent = nth root)

**Worked example:** Simplify $\frac{(4x^3 y^{-2})^3}{2x^2 y^4}$ 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. $$\frac{32x^7}{y^{10}}$$

> **tip**
>
> Always check that your final answer uses only positive exponents unless the question explicitly says otherwise.

> **Exam tip:** Double-check the sign of exponents after simplifying negative exponents, this is a common marking point.

## Exponential and Logarithmic Conversion

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

*Notation:* $\log_a x$

*Example:* $\log_3 9 = 2$, 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 $\log_4 64 = 3$ in exponential form.

1. For (a): Identify $a=5$, $b=3$, $c=125$. Substitute into the conversion rule $\log_a c = b$:
2. Result: $\log_5 125 = 3$
3. For (b): Identify $a=4$, $b=3$, $c=64$. Substitute into the conversion rule $a^b = c$:
4. Result: $4^3 = 64$

> **note**
>
> The argument of a logarithm can never be zero or negative for real-valued results. This is an important check when solving equations later.

## Laws of Logarithms

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: $\log_a (xy) = \log_a x + \log_a y$
- Quotient rule: $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$
- Power rule: $\log_a (x^n) = n \log_a x$
- Change of base rule: $\log_a b = \frac{\log_c b}{\log_c a}$ for any $c>0, c \neq 1$
- Special cases: $\log_a 1 = 0$, $\log_a a = 1$, $\log_a (1/x) = -\log_a x$

**Worked example:** Simplify $3 \log_2 6 - \log_2 27$ 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 $\log_2 8$: since $2^3 = 8$, the final result is 3.

> **mnemonic**
>
> Multiplication inside becomes addition outside, division inside becomes subtraction outside, a power inside comes outside as a multiplier.

## Common pitfalls

- **Wrong:** $(a^m)^n = a^{m^n}$ (raising $m$ to the power $n$ instead of multiplying)
  - Why it fails: Confusion between the power of a power rule and standard order of operations
  - Correct: $(a^m)^n = a^{m \times n}$
- **Wrong:** $a^m + a^n = a^{m+n}$
  - Why it fails: The exponent addition rule only applies to products of terms with the same base, not sums
  - Correct: $a^m \times a^n = a^{m+n}$; $a^m + a^n$ cannot be simplified to a single power of $a$
- **Wrong:** $\log_a (x + y) = \log_a x + \log_a y$
  - Why it fails: Incorrectly distributing the logarithm over addition; the product rule only applies to products
  - Correct: $\log_a (xy) = \log_a x + \log_a y$; there is no general simplification for $\log_a(x+y)$
- **Wrong:** $\frac{\log_a x}{\log_a y} = \log_a \left(\frac{x}{y}\right)$
  - Why it fails: Confusion between the quotient rule (for a quotient inside the log) and the change of base rule
  - Correct: $\log_a (x/y) = \log_a x - \log_a y$; $\frac{\log_a x}{\log_a y} = \log_y x$
- **Wrong:** $-2^2 = 4$
  - Why it fails: Order of operations error: the exponent applies only to the base, not the negative sign unless enclosed in parentheses
  - Correct: $-2^2 = -(2^2) = -4$, and $(-2)^2 = 4$

## Cheatsheet

| Rule Type | Exponent Form | Logarithm Form |
| --- | --- | --- |
| Product | $a^m a^n = a^{m+n}$ | $\log_a(xy) = \log_a x + \log_a y$ |
| Quotient | $a^m/a^n = a^{m-n}$ | $\log_a(x/y) = \log_a x - \log_a y$ |
| Power | $(a^m)^n = a^{mn}$ | $\log_a x^n = n \log_a x$ |
| Zero Rule | $a^0 = 1 \ (a \neq 0)$ | $\log_a 1 = 0$ |
| Identity | $a^1 = a$ | $\log_a a = 1$ |
| Negative | $a^{-n} = 1/a^n$ | $\log_a(1/x) = -\log_a x$ |
| Special | $a^{m/n} = \sqrt[n]{a^m}$ | $\log_a b = \frac{\log b}{\log a}$ (Change of Base) |

## 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 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.

- [Solving exponential and logarithmic equations](https://www.owlsprep.com/study/ib-math-aa-sl-u1-solving-exponential-and-logarithmic-equations/)
- [Binomial expansion and binomial theorem](https://www.owlsprep.com/study/ib-math-aa-sl-u1-binomial-expansion-and-binomial-theorem/)

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