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Exponents & Logarithms SL+HL

IB Mathematics: Applications & Interpretation · Topic 1: Number & Algebra
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1
Laws of Exponents
These rules apply for all real bases (\( a, b \) > 0) and all integer or rational exponents.
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
\( a^{-n} = 1 / a^n \)
(0, 1) y = 2x y = 2-x x y y = 0 asymptote
Rational (fractional) exponents
\( a^{1/n} = ^n\sqrt{}a a^{m/n} = (^n\sqrt{}a)^m \)
✗
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
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 \( e) \)
\( \ln x = log_e x \)
\( e \approx 2.71828... \)
Key relationships
\( log_a a = 1 log_a 1 = 0 log_a a^x = x a^{log_a x} = x \)
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Think: "\( \log_{a} \) \( b = x \)" means "what power of \( a \) gives \( b \)?" So log2 8 = 3 because 23 = 8.
3
Laws of Logarithms HL
Product rule
\( log_a (xy) = log_a x + log_a y \)
Quotient rule
\( log_a (x/y) = log_a x - log_a y \)
Power rule
\( log_a x^n = n log_a x \)
Change of base
\( log_a x = \log x / \log a = \ln x / \ln a \)
Worked Example
Simplify 2 log 3 + log 4 \( - \) log 6.
= log 32 + log 4 \( - \) log 6   (power rule)
= log 9 + log 4 \( - \) log 6   = log (9 \( \times 4 / 6) \)   (product & quotient rules)
= log 6
Answer: log 6
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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.
JMaths · www.jmaths.xyz Exponents & Logarithms · Page 1 of 2

Exponents & Logarithms SL+HL

IB Mathematics: Applications & Interpretation · Topic 1: Number & Algebra
www.jmaths.xyz
4
Solving Exponential Equations
To solve \( a^{x} = b, \) take logarithms of both sides or use the GDC.
Method: Take logs of both sides
\( a^x = b \to x \log a = \log b \to x = \log b / \log a \)
Worked Example
Solve 5\( x \) = 200.
\( x \) log 5 = log 200
\( x = \) log 200 / log 5 = 2.30103... / 0.69897... = 3.292...
Answer: \( x = 3.29 (3 \) s.f.)
Worked Example — Exponential model
A population is modelled by \( P = 500 \times e \)0.03\( t \). Find \( t \) when \( P = 750 \).
750 = 500 \( e \)0.03\( t \)  \( \to \)  1.5 = \( e \)0.03\( t \)
ln 1.5 = 0.03\( t \)  \( \to \)  \( t = \) ln 1.5 / 0.03 = 13.5 (3 s.f.)
Answer: \( t = 13.5 \)
GDC: Logarithms & Solving
TI-84 Plus CE
[LOG] for log10,   [LN] for ln
Change of base: [MATH] → logBASE( → logBASE(8, 2) gives 3
Solve: [MATH] → Solver → enter equation 0 = 5^X \( - 200, \) solve for X
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 Calculator app
Casio fx-CG50
[log] for log10,   [ln] for ln
Change of base: [SHIFT][log] → \( \log_{a} \)(\( b) \) template
Solve: [MENU] → Equation → Solver → 5^X - 200 = 0 \)
5
Logarithmic Scales & Applications
Logarithmic scales are used when values span many orders of magnitude. Each step on the scale represents multiplication, not addition.
pH scale
\( pH = -log_{10} [H^{+}] \)
Each pH unit = \( \times 10 \) change in [H+]
Decibels
\( L = 10 log_{10} (I / I_0) \)
\( I_0 = 10^{-12} \) W m\( -2 \) (threshold of hearing)
Richter scale (earthquake magnitude)
\( M = log_{10} (A / A_0) \)
Each whole number increase = \( \times 10 \) amplitude, \( \times 31.6 \) energy
0 7 14 ACID NEUTRAL BASE each pH unit = x10 change in [H+]
Worked Example
A sound has intensity \( I = 3.5 \times 10^{-4} \) W m\( -2 \). Find the decibel level.
\( L = 10 \) log10 (3.5 \( \times 10^{-4} / 10^{-12}) \)
= 10 log10 (3.5 \( \times 10^8) = 10 \times 8.544 = 85.4 \) dB
Answer: 85.4 dB (3 s.f.)
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IB Exam tip: Log scale questions often ask "how many times greater?" If two earthquakes differ by 2 on the Richter scale, one has 102 = 100 times the amplitude.
6
Exam Traps & Key Reminders
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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.
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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).
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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.
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Formula booklet: The laws of exponents and logarithms are given. 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 2 of 2