Numbers, Powers and Logarithms

Formulas (2)

• Prime factorisation, highest common factor and lowest common multiple · HCF and LCM of two numbers
  $$\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$$
• Fractions, recurring decimals and putting numbers in order · Order for 0 < x < 1
  $$0 < x^2 < x < \sqrt{x} < 1 < \frac{1}{\sqrt{x}} < \frac{1}{x}$$

Reference tables (1)

Watch out for (3)

• 22/7 is not π
  → The number sets ℕ, ℤ, ℚ and ℝ
• HCF uses the lowest powers, LCM the highest
  → Prime factorisation, highest common factor and lowest common multiple
• Squaring does not always make a number bigger
  → Fractions, recurring decimals and putting numbers in order

Formulas (3)

• Laws of powers and rewriting to a common base · Laws of powers
  $$a^m a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \qquad (a^m)^n = a^{mn}$$
• Fractional exponents and nth roots · Fractional exponent
  $$a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$$
• Simplifying surds and rationalising the denominator · Rationalising with the conjugate
  $$\frac{a}{b+\sqrt{c}} = \frac{a\,(b-\sqrt{c})}{b^2 - c}$$

Watch out for (4)

• Adding equal powers does not add the exponents
  → Laws of powers and rewriting to a common base
• A negative exponent does not make the number negative
  → Fractional exponents and nth roots
• The root of a sum is not the sum of the roots
  → Simplifying surds and rationalising the denominator
• The conjugate denominator is b² minus c
  → Simplifying surds and rationalising the denominator

Formulas (4)

• Scientific notation and orders of magnitude · Scientific notation
  $$x = a \times 10^{n}, \qquad 1 \le a < 10, \quad n \in \mathbb{Z}$$
• Logarithm: definition, log₁₀ and ln · Definition of the logarithm
  $$\log_b x = y \quad\text{means}\quad b^{y} = x$$
• Laws of logarithms and change of base · Laws of logarithms
  $$\log(xy) = \log x + \log y \qquad \log\frac{x}{y} = \log x - \log y \qquad \log x^{k} = k\log x \qquad \log_b x = \frac{\log x}{\log b}$$
• Solving exponential equations and rearranging log relations · Exponential equation by logs
  $$a^{x} = c \;\Rightarrow\; x = \frac{\log c}{\log a} \qquad (c > 0)$$

Watch out for (4)

• 15 × 10⁴ is not in scientific notation
  → Scientific notation and orders of magnitude
• A negative logarithm is fine; a negative argument is not
  → Logarithm: definition, log₁₀ and ln
• The log of a sum does not split
  → Laws of logarithms and change of base
• The constant in a log relation becomes a factor
  → Solving exponential equations and rearranging log relations

Formulas (3)

• Absolute value as distance from zero · Absolute value
  $$|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}$$
• Percentages: change, reverse percentage and successive changes · Percentage change as a multiplier
  $$\text{new} = \text{original} \times \left(1 \pm \frac{p}{100}\right)$$
• Ratios and sharing in a given ratio · Sharing in the ratio a : b
  $$\text{parts} = \frac{a}{a+b}\,T \quad\text{and}\quad \frac{b}{a+b}\,T$$

Watch out for (3)

• Minus x is not always negative
  → Absolute value as distance from zero
• A rise and a fall of the same percentage do not cancel
  → Percentages: change, reverse percentage and successive changes
• Adding the same amount to both parts changes the ratio
  → Ratios and sharing in a given ratio