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Exponents & Logarithms SL+HL

IB Mathematics: Applications & Interpretation · Topic 1: Number & Algebra
www.jmaths.xyz
1
Laws of Exponents
These rules apply for non-zero real bases (\( a, b \neq 0 \)) and all integer exponents (SL 1.5). For fractional exponents take \( a, b > 0 \) (HL 1.10).
Multiplication
\( a^m \times a^n = a^{m+n} \)
Division
\( a^m \div a^n = a^{m-n} \)
Power of a power
\( (a^m)^n = a^{mn} \)
Power of a product
\( (ab)^n = a^n b^n \)
Zero exponent
\( a^0 = 1 \)
\( (a \neq 0) \)
Negative exponent
\( a^{-n} = \dfrac{1}{a^n} \)
(0, 1) y = 2x y = 2-x x y y = 0 asymptote
Rational (fractional) exponents HL
\( a^{1/n} = \sqrt[n]{a}, \quad a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m} \)
Worked Example HL
Evaluate \( 27^{2/3} \), \( 16^{-3/4} \), and simplify \( \dfrac{x^{1/2}\,x^{3/2}}{x} \).
\( 27^{2/3} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9 \)
\( 16^{-3/4} = \dfrac{1}{\left(\sqrt[4]{16}\right)^3} = \dfrac{1}{2^3} = \dfrac{1}{8} \)
\( \dfrac{x^{1/2}\,x^{3/2}}{x} = \dfrac{x^{2}}{x^{1}} = x^{1} = x \)
Answers: \( 9 \), \( \tfrac{1}{8} \), \( x \)
✗
Common error: \( (a + b)^2 \neq a^2 + b^2 \). The exponent law \( (ab)^{n} = a^{n}b^{n} \) only works for products, not sums.
2
Standard (Scientific) Form SL
Standard form writes a number as \( a \times 10^{k} \) with \( 1 \leq a < 10 \) and \( k \in \mathbb{Z} \) (SL 1.1). It underpins log-scale work, where quantities span many orders of magnitude.
Definition
\( a \times 10^{k}, \quad 1 \leq a < 10, \quad k \in \mathbb{Z} \)
Multiplying / dividing
\( (a\times10^{m})(b\times10^{n}) = ab\times10^{m+n} \)
Then re-normalise \( a \) back into \( [1, 10) \).
Worked Example
Write \( 34\,500 \) and \( 0.0067 \) in standard form; evaluate \( (4\times10^{5})(8\times10^{-2}) \).
\( 34\,500 = 3.45 \times 10^{4} \),   \( 0.0067 = 6.7 \times 10^{-3} \)
\( (4\times10^{5})(8\times10^{-2}) = 32\times10^{3} = 3.2\times10^{4} \) (re-normalised)
Answer: \( 3.2\times10^{4} \)
TI-84 Plus CE
Enter standard form with [2nd][,] (the EE key): type 3.45E4
Display: [MODE] → SCI to show answers in standard form
TI-Nspire CX II
Enter with the [EE] key (or type 3.45×10^4)
Display: [doc] → Settings → Document Settings → Exponential Format = Scientific
Casio fx-CG50
Enter with the [EXP] / [×10x] key: type 3.45 EXP 4
Display: [SHIFT][MENU] (SET UP) → Display = Sci
JMaths · www.jmaths.xyz Exponents & Logarithms · Page 1 of 4

Exponents & Logarithms SL+HL

IB Mathematics: Applications & Interpretation · Topic 1: Number & Algebra
www.jmaths.xyz
3
Introduction to Logarithms
A logarithm is the inverse of an exponential. If \( a^{x} = b \), then \( \log_{a} b = x \).
Common logarithm (base 10)
\( \log x = \log_{10} x \)
Used for pH, decibels, Richter scale
Natural logarithm (base \( e \))
\( \ln x = \log_{e} x \)
\( e \approx 2.71828\ldots \)
Key relationships
\( \log_a a = 1, \quad \log_a 1 = 0, \quad \log_a a^x = x, \quad a^{\log_a x} = x \)
▶
Think: "\( \log_{a} b = x \)" means "what power of \( a \) gives \( b \)?" So \( \log_{2} 8 = 3 \) because \( 2^{3} = 8 \).
4
Laws of Logarithms HL
Combining or expanding logarithms uses the laws below (HL 1.9). Numerically evaluating a single \( \log_{10} \) or \( \ln \) is SL (1.5).
Product rule
\( \log_a (xy) = \log_a x + \log_a y \)
Quotient rule
\( \log_a \left(\dfrac{x}{y}\right) = \log_a x - \log_a y \)
Power rule
\( \log_a x^n = n \log_a x \)
Change of base
\( \log_a x = \dfrac{\log x}{\log a} = \dfrac{\ln x}{\ln a} \)
Worked Example
Simplify \( 2\log 3 + \log 4 - \log 6 \).
\( = \log 3^{2} + \log 4 - \log 6 \)   (power rule)
\( = \log 9 + \log 4 - \log 6 = \log\!\left(\dfrac{9 \times 4}{6}\right) \)   (product & quotient rules)
\( = \log 6 \)
Answer: \( \log 6 \)
✗
Common error: \( \log(x + y) \neq \log x + \log y \). The product rule is \( \log(xy) = \log x + \log y \). There is no rule for the log of a sum.
5
Solving Exponential Equations
In AI, solve \( a^{x} = b \) with the GDC first (graph intersection or a solver); use logarithms as a check or for an exact form.
GDC method (lead)
Graph \( y = a^x \) and \( y = b \); find intersection.
Or use a built-in solver for \( a^x = b \).
Algebraic check (take logs)
\( a^x = b \;\Rightarrow\; x \log a = \log b \;\Rightarrow\; x = \dfrac{\log b}{\log a} \)
Worked Example
Solve \( 5^{x} = 200 \).
GDC: intersect \( y = 5^x \) with \( y = 200 \) (or solve(5^x=200, x)) \( \to x = 3.29 \)
Check (algebra): \( x \log 5 = \log 200 \Rightarrow x = \dfrac{\log 200}{\log 5} = \dfrac{2.30103\ldots}{0.69897\ldots} = 3.292\ldots \)
Answer: \( x = 3.29 \) (3 s.f.)
Worked Example — Exponential model
A population is modelled by \( P = 500 \, e^{0.03t} \). Find \( t \) when \( P = 750 \).
\( 750 = 500 \, e^{0.03t} \;\to\; 1.5 = e^{0.03t} \)
\( \ln 1.5 = 0.03t \;\to\; t = \dfrac{\ln 1.5}{0.03} = 13.5 \) (3 s.f.)
Answer: \( t = 13.5 \)
JMaths · www.jmaths.xyz Exponents & Logarithms · Page 2 of 4

Exponents & Logarithms SL+HL

IB Mathematics: Applications & Interpretation · Topic 1: Number & Algebra
www.jmaths.xyz
5b
GDC: Logarithms & Solving
TI-84 Plus CE
[LOG] for log10,   [LN] for ln
Change of base: [MATH] → logBASE( → logBASE(8, 2) gives 3
Solve: graph \( y = 5^X \) and \( y = 200 \), then [2nd][TRACE] → intersect; or [MATH] → Solver → enter \( 0 = 5\hat{}X - 200 \)
TI-Nspire CX II
Type log(x) for base 10,   ln(x) for base \( e \)
For other bases: use the log template [log□] or type log(8,2)
Solve: type solve(5^x = 200, x) in the Calculator app, or use a Graphs page intersection
Casio fx-CG50
[log] for log10,   [ln] for ln
Change of base: in Run-Matrix, open the [MATH] keyboard menu → \( \log_{\square}(\square) \) template (enter base, then argument). Not [SHIFT][log] — that gives \( 10^{x} \).
Solve: [MENU] → Equation → Solver → \( 5\hat{}X - 200 = 0 \); or Graph two curves and use G-Solve intersection
6
Logarithmic Scales & Applications
Logarithmic scales are used when values span many orders of magnitude. Each step represents multiplication, not addition. Evaluating a given log-scale formula is SL-accessible (the formula is supplied in the question; you substitute and solve with \( \log_{10} \)). Manipulating these with the laws of logs is HL (1.9).
pH scale
\( \mathrm{pH} = -\log_{10} [\mathrm{H}^{+}] \)
Each \( +1 \) pH unit \( = [\mathrm{H}^{+}] \) divided by 10.
Decibels
\( L = 10 \log_{10} \left(\dfrac{I}{I_0}\right) \)
\( I_0 = 10^{-12} \) W m\( -2 \) (threshold of hearing)
Richter scale (earthquake magnitude)
\( M = \log_{10} \left(\dfrac{A}{A_0}\right) \)
Each whole number increase \( = \times 10 \) amplitude (\( \times 31.6 \) energy — enrichment). The AI exam supplies the Gutenberg–Richter form \( \log_{10} N = a - M \).
0 7 14 ACID NEUTRAL BASE each +1 pH unit = [H+] divided by 10
Worked Example
A sound has intensity \( I = 3.5 \times 10^{-4} \) W m\( -2 \). Find the decibel level.
\( L = 10 \log_{10} \left(\dfrac{3.5 \times 10^{-4}}{10^{-12}}\right) \)
\( = 10 \log_{10} (3.5 \times 10^{8}) = 10 \times 8.544 = 85.4 \) dB
Answer: 85.4 dB (3 s.f.)
▶
IB Exam tip: Log scale questions often ask "how many times greater?" If two earthquakes differ by 2 on the Richter scale, one has \( 10^{2} = 100 \) times the amplitude.
JMaths · www.jmaths.xyz Exponents & Logarithms · Page 3 of 4

Exponents & Logarithms SL+HL

IB Mathematics: Applications & Interpretation · Topic 1: Number & Algebra
www.jmaths.xyz
7
Linearizing Data with Logarithms HL
A heavily-examined HL pattern (2.10): take logs of tabulated data so a curved relationship becomes a straight line, then run linear regression on the GDC.
Log-log (power model)
\( y = a x^{n} \;\Rightarrow\; \log y = \log a + n \log x \)
Plot \( \log y \) vs \( \log x \): gradient \( = n \), intercept \( = \log a \).
Semi-log (exponential model)
\( y = k a^{x} \;\Rightarrow\; \log y = \log k + x \log a \)
Plot \( \log y \) vs \( x \): gradient \( = \log a \), intercept \( = \log k \).
✓ Method: take logs of the listed data → run linear regression on the GDC → read gradient/intercept → back-solve \( a, n \) (or \( k, a \)). Give the final answer in the original \( y = \ldots \) form ("answer should not include logarithms").
Worked Example HL
A log-log line passes through \( (\log 2, \log 13.20) \) and \( (\log 4, \log 34.82) \). Find the power model \( y = a x^{n} \).
\( n = \dfrac{\log 34.82 - \log 13.20}{\log 4 - \log 2} = \dfrac{0.5419 - 0.1206}{0.6021 - 0.3010} \approx 1.40 \)
\( \log a = \log 13.20 - 1.40 \log 2 = 0.1206 - 1.40(0.3010) \approx -0.301 \Rightarrow a \approx 5.0 \)
Answer: \( y \approx 5\,x^{1.40} \)
TI-84 Plus CE
Put \( \log x \) in L1 and \( \log y \) in L2 (or \( x \) and \( \log y \) for semi-log)
[STAT] → CALC → LinReg(ax+b) on L1, L2: gradient \( a \), intercept \( b \)
TI-Nspire CX II
Enter transformed data in Lists & Spreadsheet
[menu] → Statistics → Stat Calculations → Linear Regression (mx+b): read \( m \) and \( b \)
Casio fx-CG50
[MENU] → Statistics; enter \( \log x \), \( \log y \) in lists
[CALC] → [REG] → X (linear): read gradient and intercept
8
Exam Traps & Key Reminders
✗
Negative under the log. \( \log x \) and \( \ln x \) are only defined for \( x > 0 \). If a question gives \( x \leq 0 \), the answer is "no solution" or restrict the domain.
✗
log vs ln confusion. Use \( \ln \) when the equation involves \( e \); use \( \log \) when it involves powers of 10. For solving \( a^{x} = b \) with any base, either works (change of base).
✗
Forgetting to check solutions. After solving a log equation, always substitute back to check no argument is negative. e.g. \( \log(x - 3) = 2 \) gives \( x = 103 \), which is valid since \( 103 - 3 > 0 \).
▶
Formula booklet: The laws of exponents and logarithms are given. The change of base formula is given. The pH and decibel formulas may be given in context within the question.
JMaths · www.jmaths.xyz Exponents & Logarithms · Page 4 of 4