Exponents and logarithms

IB Mathematics AA HL· Unit 1: Number & Algebra, Topic 3· 15 min read

1. Laws of Exponents★★☆☆☆⏱ 8 min

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📘 Definition

Exponential Expression

$a^{n}$

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

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:

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

📐 Worked Example

Simplify $\frac{( 8 x ^{3} y ^{- 2} ) ^{\frac{1}{3}}}{( 4 x ^{4} y ^{- 3} ) ^{- \frac{1}{2}}}$ into the form $k x^{a} y^{b}$

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.

2. Laws of Logarithms★★☆☆☆⏱ 10 min

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📘 Definition

Logarithm

$lo g_{a} b$

For $a > 0, a \neq = 1, b > 0$ , $lo g_{a} b = c$ if and only if $a^{c} = b$ . Logarithms are the inverse function of exponential functions with base $a$ .

Example:

$lo g_{2} 8 = 3$ because $2^{3} = 8$

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

$lo g_{a} (x y) = lo g_{a} x + lo g_{a} y$
$lo g_{a} (\frac{x}{y}) = lo g_{a} x - lo g_{a} y$
$lo g_{a} (x^{n}) = n lo g_{a} x$
$lo g_{a} 1 = 0$ , $lo g_{a} a = 1$
$a^{l o g_{a} b} = b$ , $lo g_{a} a^{c} = c$

📐 Worked Example

Express $2 lo g_{3} 6 - \frac{1}{2} lo g_{3} 16 + 3 lo g_{3} 2$ as a single logarithm

1
Apply the power rule $n lo g_{a} x = lo g_{a} x^{n}$ 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
$\log_3 \left(\frac{36 \times 8}{4}\right) = \log_3 72$

3. Change of Base Formula★★★☆☆⏱ 7 min

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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).

📘 Definition

Change of Base Formula

$lo g_{a} c$

For $a > 0, a \neq = 1$ and $b > 0, b \neq = 1$ , $lo g_{a} c = \frac{l o g _{b} c}{l o g _{b} a}$ . The most common form uses natural logs: $lo g_{a} c = \frac{l n c}{l n a}$ .

📐 Worked Example

Find the exact value of $lo g_{4} 8 + lo g_{2} 12$

1
Rewrite $lo g_{4} 8$ 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
$\frac{3}{2} + \log_2 12 = \log_2 2^{3/2} + \log_2 12 = \log_2 (12 \times 2\sqrt{2}) = \log_2 (24\sqrt{2})$

4. Solving Exponential and Logarithmic Equations★★★☆☆⏱ 12 min

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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 $2^{2 x + 1} = 3^{x - 2}$ , 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 $lo g_{2} (x + 3) + lo g_{2} (x - 1) = 3$

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: $x = - 1 + 23 \approx 2.46$

5. Common Pitfalls

Wrong move:

Expanding $lo g_{a} (x + y)$ as $lo g_{a} x + lo g_{a} y$

Why:

The logarithm product rule only applies to products, not sums of arguments

Correct move:

Only apply $lo g_{a} (x y) = lo g_{a} x + lo g_{a} y$ to products, never sums

Wrong move:

Keeping negative solutions that make any logarithm argument non-positive

Why:

Logarithms are undefined for non-positive arguments, so these solutions are extraneous

Correct move:

Always check all solutions against the domain of the original equation

Wrong move:

Rewriting $(ln x)^{2}$ as $2 ln x$

Why:

$(ln x)^{2} = ln x \times ln x$ , which is not the same as $ln (x^{2})$

Correct move:

Only use $n ln x = ln x^{n}$ for the logarithm of a power, not a power of a logarithm

Wrong move:

Simplifying $a^{m^{n}}$ as $a^{mn}$

Why:

Exponentiation is right-associative, so $a^{m^{n}} = a^{(m^{n})}$ not $(a^{m})^{n}$

Correct move:

Evaluate the top exponent first when working with stacked exponents

Wrong move:

Simplifying $\frac{l n 5}{l n 2}$ as $ln 5 - ln 2$

Why:

The fraction of logarithms from change of base is not the logarithm of a fraction

Correct move:

Remember change of base is $lo g_{a} b = \frac{l n b}{l n a}$ , not $ln (\frac{b}{a})$

6. Quick Reference Cheatsheet

Rule Type	Formula
Exponent Product	$a^{m} a^{n} = a^{m + n}$
Exponent Power	$(a^{m})^{n} = a^{mn}$
Log Product	$lo g_{a} (x y) = lo g_{a} x + lo g_{a} y$
Log Power	$lo g_{a} (x^{n}) = n lo g_{a} x$
Inverse Identity	$a^{l o g_{a} b} = b$ , $lo g_{a} a^{c} = c$
Change of Base	$lo g_{a} c = \frac{l n c}{l n a}$
Definition	$lo g_{a} b = c ⟺ a^{c} = b$
Log Quotient	$lo g_{a} \frac{x}{y} = lo g_{a} x - lo g_{a} y$

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.

2023 · Paper 1
Simplify mixed logarithmic expression
2022 · Paper 2
Solve exponential equation for x
2021 · Paper 1
Apply change of base formula

Going deeper

syllabusIB AA HL Syllabus Unit 1Official syllabus reference

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.