# Exponents and logarithms

> IB Mathematics AA HL · Number & Algebra
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u1-exponents-and-logarithms/

This module covers the laws of exponents and logarithms, their inverse relationship, the change of base formula, and solving exponential/logarithmic equations, core content for both IB AA HL Papers 1 and 2.

**Prerequisites:** [Basic index notation](https://www.owlsprep.com/study/ib-math-aa-hl-u1-basic-indices/); [Inverse function properties](https://www.owlsprep.com/study/ib-math-aa-hl-u2-inverse-functions/)

## Learning objectives

- Simplify expressions using laws of exponents and logarithms
- Explain the inverse relationship between exponential and logarithmic functions
- Apply the change of base formula for logarithms
- Solve exponential and logarithmic equations, checking for extraneous solutions

## Laws of Exponents

**Exponential Expression** — An expression representing repeated multiplication of the base $a$, extended to all real exponents for positive bases

*Notation:* $a^n$

*Example:* $2^3 = 2 \times 2 \times 2 = 8$

For any positive real numbers $a, b$ and real numbers $m, n$, the following core laws hold:

- $a^m \times a^n = a^{m+n}$
- $a^m \div a^n = a^{m-n}$
- $(a^m)^n = a^{mn}$
- $(ab)^n = a^n b^n$
- $a^0 = 1$, $a^{-n} = \frac{1}{a^n}$
- $a^{\frac{1}{n}} = \sqrt[n]{a}$

**Worked example:** Simplify $\frac{(8x^3 y^{-2})^{\frac{1}{3}}}{(4x^4 y^{-3})^{-\frac{1}{2}}}$ into the form $kx^a y^b$

1. Apply the power of a product rule to numerator and denominator:
2. $$\frac{8^{1/3} (x^3)^{1/3} (y^{-2})^{1/3}}{4^{-1/2} (x^4)^{-1/2} (y^{-3})^{-1/2}}$$
3. Simplify exponents using the power of a power rule $(a^m)^n = a^{mn}$:
4. $$\frac{2 x^1 y^{-2/3}}{\frac{1}{2} x^{-2} y^{3/2}}$$
5. Combine terms using exponent division rules:
6. $$(2 \div \frac{1}{2}) \times x^{1 - (-2)} \times y^{-\frac{2}{3} - \frac{3}{2}} = 4x^3 y^{-\frac{13}{6}}$$

> **Exam tip:** IB accepts both negative exponents and fractional forms unless the question explicitly specifies one format.

*Calculator:* forbidden

## Laws of Logarithms

**Logarithm** — For $a>0, a \neq 1, b>0$, $\log_a b = c$ if and only if $a^c = b$. Logarithms are the inverse function of exponential functions with base $a$.

*Notation:* $\log_a b$

*Example:* $\log_2 8 = 3$ because $2^3 = 8$

All logarithm laws are derived from exponent laws, and give us the following rules for $a>0, a \neq 1, x>0, y>0$:

- $\log_a (xy) = \log_a x + \log_a y$
- $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$
- $\log_a (x^n) = n \log_a x$
- $\log_a 1 = 0$, $\log_a a = 1$
- $a^{\log_a b} = b$, $\log_a a^c = c$

**Worked example:** Express $2 \log_3 6 - \frac{1}{2} \log_3 16 + 3 \log_3 2$ as a single logarithm

1. Apply the power rule $n \log_a x = \log_a x^n$ to each term:
2. $$\log_3 6^2 - \log_3 16^{1/2} + \log_3 2^3$$
3. Simplify the powers inside the logarithms:
4. $$\log_3 36 - \log_3 4 + \log_3 8$$
5. Combine using product and quotient logarithm rules:
6. $$\log_3 \left(\frac{36 \times 8}{4}\right) = \log_3 72$$

*Calculator:* forbidden

## Change of Base Formula

When working with logarithms of different bases, or calculating decimal values for non-standard bases, we use the change of base formula to convert to a common base (usually $e$ or 10).

**Change of Base Formula** — For $a>0, a \neq 1$ and $b>0, b \neq 1$, $\log_a c = \frac{\log_b c}{\log_b a}$. The most common form uses natural logs: $\log_a c = \frac{\ln c}{\ln a}$.

*Notation:* $\log_a c$

**Worked example:** Find the exact value of $\log_4 8 + \log_2 12$

1. Rewrite $\log_4 8$ with base 2 using change of base:
2. $$\log_4 8 = \frac{\log_2 8}{\log_2 4} = \frac{3}{2}$$
3. Add the two terms and simplify using logarithm rules:
4. $$\frac{3}{2} + \log_2 12 = \log_2 2^{3/2} + \log_2 12 = \log_2 (12 \times 2\sqrt{2}) = \log_2 (24\sqrt{2})$$

> **tip**
>
> Always convert all logarithms to the same base when combining or simplifying expressions with multiple bases.

*Calculator:* allowed

## Solving Exponential and Logarithmic Equations

We use the inverse relationship between exponents and logarithms to solve both types of equations. A critical final step is always checking for extraneous solutions, since logarithms are only defined for positive arguments.

**Worked example:** Solve $2^{2x+1} = 3^{x-2}$, give answer to 3 significant figures

1. Take the natural logarithm of both sides:
2. $$\ln(2^{2x+1}) = \ln(3^{x-2})$$
3. Apply the power rule of logarithms:
4. $$(2x+1)\ln 2 = (x-2)\ln 3$$
5. Rearrange to collect terms in $x$ on the left-hand side:
6. $$2x \ln 2 - x \ln 3 = -2 \ln 3 - \ln 2$$
7. Factor and solve for $x$:
8. $$x = -\frac{2 \ln 3 + \ln 2}{2 \ln 2 - \ln 3} \approx -30.7$$

**Worked example:** Solve $\log_2 (x+3) + \log_2 (x-1) = 3$

1. Combine logarithms using the product rule:
2. $$\log_2 [(x+3)(x-1)] = 3$$
3. Rewrite in exponential form:
4. $$(x+3)(x-1) = 2^3 = 8$$
5. Rearrange to form a quadratic equation:
6. $$x^2 + 2x - 11 = 0$$
7. Solve the quadratic:
8. $$x = -1 \pm 2\sqrt{3}$$
9. Check domain: logarithms require $x>1$, so discard the negative solution:
10. Final solution: $x = -1 + 2\sqrt{3} \approx 2.46$

> **warning**
>
> IB examiners explicitly award marks for checking and discarding extraneous solutions, so never skip this step.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Expanding $\log_a(x+y)$ as $\log_a x + \log_a y$
  - Why it fails: The logarithm product rule only applies to products, not sums of arguments
  - Correct: Only apply $\log_a(xy) = \log_a x + \log_a y$ to products, never sums
- **Wrong:** Keeping negative solutions that make any logarithm argument non-positive
  - Why it fails: Logarithms are undefined for non-positive arguments, so these solutions are extraneous
  - Correct: Always check all solutions against the domain of the original equation
- **Wrong:** Rewriting $(\ln x)^2$ as $2 \ln x$
  - Why it fails: $(\ln x)^2 = \ln x \times \ln x$, which is not the same as $\ln(x^2)$
  - Correct: Only use $n \ln x = \ln x^n$ for the logarithm of a power, not a power of a logarithm
- **Wrong:** Simplifying $a^{m^n}$ as $a^{mn}$
  - Why it fails: Exponentiation is right-associative, so $a^{m^n} = a^{(m^n)}$ not $(a^m)^n$
  - Correct: Evaluate the top exponent first when working with stacked exponents
- **Wrong:** Simplifying $\frac{\ln 5}{\ln 2}$ as $\ln 5 - \ln 2$
  - Why it fails: The fraction of logarithms from change of base is not the logarithm of a fraction
  - Correct: Remember change of base is $\log_a b = \frac{\ln b}{\ln a}$, not $\ln(\frac{b}{a})$

## Cheatsheet

| Rule Type | Formula |
| --- | --- |
| Exponent Product | $a^m a^n = a^{m+n}$ |
| Exponent Power | $(a^m)^n = a^{mn}$ |
| Log Product | $\log_a(xy) = \log_a x + \log_a y$ |
| Log Power | $\log_a(x^n) = n \log_a x$ |
| Inverse Identity | $a^{\log_a b} = b$, $\log_a a^c = c$ |
| Change of Base | $\log_a c = \frac{\ln c}{\ln a}$ |
| Definition | $\log_a b = c \iff a^c = b$ |
| Log Quotient | $\log_a \frac{x}{y} = \log_a x - \log_a y$ |

## What's next

Exponents and logarithms are foundational to almost all other topics in IB AA HL. You will use these laws constantly when working with exponential modeling, differentiation and integration of exponential and logarithmic functions, sequences and series, and solving differential equations. Small mistakes in this topic cascade into errors in more complex problems, so mastering the rules here will save you significant marks later. Next, you will extend this knowledge to study the graphs and properties of exponential and logarithmic functions, before applying these concepts to calculus.

- [Sequences & Series](https://www.owlsprep.com/study/ib-math-aa-hl-u1-sequences-and-series/)
- [Counting Principles](https://www.owlsprep.com/study/ib-math-aa-hl-u1-counting-principles/)
- [Binomial Theorem](https://www.owlsprep.com/study/ib-math-aa-hl-u1-binomial-theorem/)

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