IB Math AA SL Topic 1.7: Exponent & log laws | Notes & Flashcards | Aimnova

IB Math AA SL — Exponent & log laws

Topic 1.7 of IB Mathematics: Analysis and Approaches covers Exponent & log laws, which is part of Unit 1: Number & Algebra. Students explore key concepts including Laws of exponents, Laws of logarithms, Exponential & log equations. A strong understanding of exponent & log laws is essential for IB Math AA SL exams and builds the foundation for connected topics across the syllabus.

Key concepts in Exponent & log laws

Key Idea: These are your algebra cheat-sheets for powers and logs — used to simplify expressions and to solve exponential and log equations. Almost all of it is Paper 1 by hand; Paper 2 only adds a graphing shortcut.

⚡ Laws of exponents (Paper 1)

a^m a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}

$a$: the common base — must be the same to combine
$a^{-n} = \tfrac{1}{a^n}$: negative power = reciprocal
$a^{m/n} = (\sqrt[n]{a})^{m}$: fractional power = nth root, then mth power

🟰 Laws of logarithms (Paper 1)

\log_a xy = \log_a x + \log_a y, \quad \log_a \tfrac{x}{y} = \log_a x - \log_a y, \quad \log_a x^m = m\log_a x, \quad \log_a x = \frac{\log_b x}{\log_b a}

$\log_a 1 = 0$: log of 1 is always 0
$\log_a a = 1$: log of the base is 1
$\frac{\log_b x}{\log_b a}$: change of base — swap to any base you can compute

✏️ IB-style worked examples

IB-style question — simplify with the index laws

Simplify (p⁵ × p³) ÷ p², leaving your answer as a single power of p.

Step by step:

Multiply — add the exponents.
$p^5 \times p^3 = p^{5+3} = p^8$
Divide — subtract the exponent.
$\frac{p^8}{p^2} = p^{8-2} = p^6$

Final answer:

p⁶

IB-style question — condense into a single logarithm

Write ln 5 + 2 ln 2 − ln 10 as a single logarithm.

Step by step:

Coefficient first: the power law moves the 2 up.
$2\ln 2 = \ln 2^2 = \ln 4$
Add → multiply on top, subtract → divide.
$\ln 5 + \ln 4 - \ln 10 = \ln \frac{5 \times 4}{10}$
Simplify the inside.
$= \ln 2$

Final answer:

ln 2

IB-style question — solve an exponential by taking logs

Solve 7ˣ = 50, giving the exact value of x.

Step by step:

Bases won't match, so take logs of both sides.
$\log 7^x = \log 50$
The power law brings x down.
$x \log 7 = \log 50$
Divide to isolate x.
$x = \frac{\log 50}{\log 7} \approx 2.01$

Final answer:

x = log 50 / log 7 (≈ 2.01)

Important: log(x + y) ≠ log x + log y. The laws act only on a product, quotient or power — never on a sum or difference inside one log. And after solving a log equation, check every argument is positive: discard any root that makes a log's inside ≤ 0.

Tap each card to reveal the answer.

Exam Tips

Same base only: multiply → add exponents, divide → subtract, power of a power → multiply.
Negative power = reciprocal; fractional power m/n = nth root then mth power.
Logs: product → add, quotient → subtract, power → coefficient (read right-to-left to combine).
Bases match → equate exponents; otherwise take logs. A lone log = number → convert to expr = aᶜ.
Reject any solution that makes a log's argument ≤ 0, and use change of base for an awkward base.

IB Math AA SL — Exponent & log laws

Key concepts in Exponent & log laws

Key Idea: These are your algebra cheat-sheets for powers and logs — used to simplify expressions and to solve exponential and log equations. Almost all of it is Paper 1 by hand; Paper 2 only adds a graphing shortcut.

⚡ Laws of exponents (Paper 1)

a^m a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}

$a$: the common base — must be the same to combine
$a^{-n} = \tfrac{1}{a^n}$: negative power = reciprocal
$a^{m/n} = (\sqrt[n]{a})^{m}$: fractional power = nth root, then mth power

🟰 Laws of logarithms (Paper 1)

\log_a xy = \log_a x + \log_a y, \quad \log_a \tfrac{x}{y} = \log_a x - \log_a y, \quad \log_a x^m = m\log_a x, \quad \log_a x = \frac{\log_b x}{\log_b a}

$\log_a 1 = 0$: log of 1 is always 0
$\log_a a = 1$: log of the base is 1
$\frac{\log_b x}{\log_b a}$: change of base — swap to any base you can compute

✏️ IB-style worked examples

IB-style question — simplify with the index laws

Simplify (p⁵ × p³) ÷ p², leaving your answer as a single power of p.

Step by step:

Multiply — add the exponents.
$p^5 \times p^3 = p^{5+3} = p^8$
Divide — subtract the exponent.
$\frac{p^8}{p^2} = p^{8-2} = p^6$

Final answer:

p⁶

IB-style question — condense into a single logarithm

Write ln 5 + 2 ln 2 − ln 10 as a single logarithm.

Step by step:

Coefficient first: the power law moves the 2 up.
$2\ln 2 = \ln 2^2 = \ln 4$
Add → multiply on top, subtract → divide.
$\ln 5 + \ln 4 - \ln 10 = \ln \frac{5 \times 4}{10}$
Simplify the inside.
$= \ln 2$

Final answer:

ln 2

IB-style question — solve an exponential by taking logs

Solve 7ˣ = 50, giving the exact value of x.

Step by step:

Bases won't match, so take logs of both sides.
$\log 7^x = \log 50$
The power law brings x down.
$x \log 7 = \log 50$
Divide to isolate x.
$x = \frac{\log 50}{\log 7} \approx 2.01$

Final answer:

x = log 50 / log 7 (≈ 2.01)

Important: log(x + y) ≠ log x + log y. The laws act only on a product, quotient or power — never on a sum or difference inside one log. And after solving a log equation, check every argument is positive: discard any root that makes a log's inside ≤ 0.

Tap each card to reveal the answer.

Exam Tips

Same base only: multiply → add exponents, divide → subtract, power of a power → multiply.
Negative power = reciprocal; fractional power m/n = nth root then mth power.
Logs: product → add, quotient → subtract, power → coefficient (read right-to-left to combine).
Bases match → equate exponents; otherwise take logs. A lone log = number → convert to expr = aᶜ.
Reject any solution that makes a log's argument ≤ 0, and use change of base for an awkward base.

IB Math AA SL — Exponent & log laws

Key concepts in Exponent & log laws

⚡ Laws of exponents (Paper 1)

🟰 Laws of logarithms (Paper 1)

✏️ IB-style worked examples

IB-style question — simplify with the index laws

IB-style question — condense into a single logarithm

IB-style question — solve an exponential by taking logs

Exam Tips

What you'll learn in Topic 1.7

Study resources — 1.7 Exponent & log laws

Laws of exponents

Laws of logarithms

Exponential & log equations

Ready to study Exponent & log laws?

Ready to practice?

IB Math AA SL — Exponent & log laws

Key concepts in Exponent & log laws

⚡ Laws of exponents (Paper 1)

🟰 Laws of logarithms (Paper 1)

✏️ IB-style worked examples

IB-style question — simplify with the index laws

IB-style question — condense into a single logarithm

IB-style question — solve an exponential by taking logs

Exam Tips

What you'll learn in Topic 1.7

Study resources — 1.7 Exponent & log laws

Laws of exponents

Laws of logarithms

Exponential & log equations

Ready to study Exponent & log laws?

Ready to practice?