IB DP Maths AI · Topic 1 Number and algebra · SL 1.5, 1.8 · AHL 1.9, 1.10 · ~6 min read

Exponents, logarithms and equations

Laws of exponents, an introduction to logarithms with base 10 and e, log scales such as pH and decibels, solving systems of linear equations and polynomial equations with the GDC, and for HL: the laws of logarithms and rational exponents.

Laws of exponentsLogarithms base 10 and eSolving aˣ = bpH and decibel scalesSystems of linear equationsPolynomial equationsLaws of logarithms (HL)Rational exponents (HL)

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No tag = SL and HLHL ONLY HL students onlyFORMULA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

Exponent laws: a^m × aⁿ = a^{m + n}, a^m ÷ aⁿ = a^{m − n}, (a^m)ⁿ = a^mn, a⁰ = 1 and a⁻ⁿ = 1aⁿ.
A logarithm is an exponent: a^x = b ⇔ x = log_a b. Your GDC has log (base 10) and ln (base e ≈ 2.718).
Equations on the GDC: systems of up to three linear equations and polynomial equations are solved with the GDC's solvers. Write the equations you entered, then the solutions.
HL: laws of logarithms, log_a(xy) = log_a x + log_a y and friends, and rational exponents such as 8^2/3 = 4.

By the end you can

Simplify expressions with integer exponents.
Use logarithms with base 10 and e to solve exponential equations, including in context.
Use the GDC to solve systems of linear equations and polynomial equations.
Use the laws of logarithms. HL
Simplify expressions with rational exponents. HL

Syllabus noteFrom first exams in 2029, logarithms leave the SL syllabus. SL students sitting exams up to November 2028 still need the logarithm section of this note; the exponent and equation sections stay.

1Laws of exponents

Law	Example
a^m × aⁿ = a^{m + n}	x³ × x⁴ = x⁷
a^m ÷ aⁿ = a^{m − n}	5⁶ ÷ 5² = 5⁴
(a^m)ⁿ = a^mn	(2³)² = 2⁶ = 64
a⁻ⁿ = 1/aⁿ	10⁻³ = 0.001
a⁰ = 1	7⁰ = 1

Worked example

Simplify (2x³)² × x⁻⁴.

(2x³)² = 4x⁶ M1
4x⁶ × x⁻⁴ = 4x² A1

Worked example

Solve 3 × 2^x = 48.

2^x = 16 M1
16 = 2⁴, so x = 4 A1

Common mistakeWriting (2x³)² = 2x⁶. The power applies to the 2 as well: 2² = 4.

2Logarithms

Definition: log₁₀ 1000 = 3 because 10³ = 1000. ln x = log_e x, so ln e⁵ = 5.

a^x = b ⇔ x = log_a b FORMULA BOOKLET

a > 0, b > 0

Worked example

Solve 10^x = 250 and e^x = 7.

x = log₁₀ 250 = 2.40 A1
x = ln 7 = 1.95 A1

Worked example

A substance decays as M = 80e^−0.03t grams after t years. When is the mass 20 g?

e^−0.03t = 0.25 M1
−0.03t = ln 0.25 ⇒ t = 46.2 years M1A1

Log scales: pH = −log₁₀[H⁺]; sound level in decibels = 10 log₁₀(II₀) with I₀ = 10⁻¹² W m⁻². A log scale squeezes huge ranges onto a short line.

Every step of 1 on the pH scale is a factor of 10 in hydrogen ion concentration.

Worked example

Find the pH of a solution with [H⁺] = 3.2 × 10⁻⁵ mol dm⁻³, and the sound level of a noise with I = 10⁻⁵ W m⁻².

pH = −log₁₀(3.2 × 10⁻⁵) = 4.49 A1
10 log₁₀(10⁻⁵10⁻¹²) = 10 × 7 = 70 dB M1A1

GDC (calculator) methodTo solve an exponential equation without logs, graph both sides (y = 80e^−0.03x and y = 20) and find the intersection. This works on every paper in AI.

3Solving equations with the GDC

Worked example

Solve 2x + y − z = 2, x − y + 2z = 7, 3x + 2y + z = 11.

enter the coefficients into the simultaneous equation solver M1
x = 2, y = 1, z = 3 A1

Worked example

A shop sells 3 coffees and 2 cakes for $19, and 1 coffee and 4 cakes for $23. Find the price of each.

3c + 2k = 19 and c + 4k = 23 M1
coffee $3, cake $5 A1

Worked example

Solve x³ − 6x² + 9x − 2 = 0.

polynomial root finder, degree 3, coefficients 1, −6, 9, −2 M1
x = 0.268, 2, 3.73 A1

Exam tipAlways write the equations you solved before the GDC answer. "From GDC: x = 2" with no equations can lose the method mark if the answer is wrong.

4Laws of logarithms HL

log_a xy = log_a x + log_a y, log_a xy = log_a x − log_a y, log_a x^m = m log_a x FORMULA BOOKLET

the three laws

log_b x = log_a xlog_a b FORMULA BOOKLET

change of base

Worked example

Evaluate log 2 + log 50, and solve log₂ x + log₂(x − 2) = 3.

log(2 × 50) = log 100 = 2 A1
log₂[x(x − 2)] = 3 ⇒ x² − 2x − 8 = 0 M1
x = 4 or −2; x = 4 (x > 2 for the logs to exist) A1R1

5Rational exponents HL

a^1/n = ⁿ√a and a^m/n = (ⁿ√a)^m. So 8^2/3 = (∛8)² = 4 and 27^−1/3 = 13.

Worked example

Simplify x^1/2 × x^3/2x⁻¹.

x^{1/2 + 3/2} = x² M1
x² ÷ x⁻¹ = x³ A1

Linking questionExponential equations are the backbone of Modelling with functions, and log scales are used to linearise data in Transformations and linearising data (HL). In Maths AA, logarithms are proved and used algebraically: see Exponents and logarithms.

Idea for your IA / explorationInvestigate a real logarithmic scale: earthquake magnitudes, decibels or pH. Collect data and show how the log scale makes comparisons easier, or model the cooling of a drink with an exponential function.

Test yourself — cover the answers first

Quick check

Simplify x⁵ ÷ x⁻².
Evaluate 4⁻².
Solve 10^x = 0.01.
What is ln e?
A solution has pH 3. What is [H⁺]?
HL Write log 5 + log 4 as a single log.

Exam-style

The value of a painting is V = 1200 × 1.08^t dollars, t years after it was bought. Find the value after 5 years and how long it takes to reach $5000.
HL Solve log₃(x + 1) − log₃ x = 2.

Show answers

x⁷
1/16
x = −2
1
10⁻³ mol dm⁻³
log 20
V(5) = 1200 × 1.08⁵ = $1763.19
1.08^t = 50001200 ⇒ t = ln(25/6)ln 1.08 = 18.5 years
log₃ x + 1x = 2 ⇒ x + 1x = 9
x + 1 = 9x
x = 18

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