Logarithms — NDA Maths

Formulas (5)

• What a Logarithm Is — Laws, Special Values, Domain · The defining equivalence and the three laws
  $$\log_a N = x \iff a^x = N;\quad \log_a(MN)=\log_a M+\log_a N,\ \ \log_a M^k = k\log_a M$$
• Applying the Laws — Evaluate and Combine · Rewrite as powers, then pull the exponent out
  $$\log_a(b^m \cdot c^n) = m\log_a b + n\log_a c$$
• Change of Base & the Reciprocal Identity · Change of base and its reciprocal twin
  $$\log_b a = \dfrac{\log a}{\log b}, \qquad \dfrac{1}{\log_a b} = \log_b a$$
• Sign of a Logarithm & Bounds of a Log Function · Sign of a log (base > 1)
  $$\log_a N \;\begin{cases}>0 & N>1\\ =0 & N=1\\ <0 & 0<N<1\end{cases}$$
• Logarithms in AP/GP and the Geometric Mean · AP and GP conditions for three terms
  $$\text{AP}: 2q = p+r, \qquad \text{GP}: q^2 = pr$$

Watch out for (7)

• $\log(M+N)$ is NOT $\log M + \log N$
  → What a Logarithm Is — Laws, Special Values, Domain
• The base must be $>0$ and $\neq 1$
  → What a Logarithm Is — Laws, Special Values, Domain
• Keep the base when you pull out a power
  → Applying the Laws — Evaluate and Combine
• Reciprocal flips the base and the argument together
  → Change of Base & the Reciprocal Identity
• Change of base does NOT invert the fraction
  → Change of Base & the Reciprocal Identity
• The minimum is of the LOG, not the quadratic
  → Sign of a Logarithm & Bounds of a Log Function
• AP holding does not make it GP — test GP separately
  → Logarithms in AP/GP and the Geometric Mean

Formulas (5)

• Taking the Log of an Exponential Equation · Bring the exponent down with a log
  $$a^x = b \;\Rightarrow\; x = \dfrac{\log b}{\log a}$$
• Substitution t = aˣ to a Quadratic · Let t = aˣ and solve the quadratic (keep t > 0)
  $$t = a^x > 0,\quad t^2 + bt + c = 0 \;\Rightarrow\; x = \log_a t$$
• Domain Checks & Counting Solutions · Keep only roots with every argument > 0
  $$\log_a M = \log_a N \Rightarrow M = N,\ \text{then require } M,N > 0$$
• GP, Chain-Rule & AM-GM Conditions · Chain rule and the AM-GM floor
  $$\log_x a\cdot\log_b x = \log_b a, \qquad t + \tfrac{1}{t} \ge 2\ (t>0)$$
• Application — Trailing Zeros of a Factorial · Legendre — trailing zeros count the 5s
  $$Z(n) = \sum_{i\ge 1}\left\lfloor \dfrac{n}{5^i} \right\rfloor$$

Watch out for (6)

• $\log_{10} 0.2 = \log_{10} 2 - 1$, which is negative
  → Taking the Log of an Exponential Equation
• Throw out the non-positive t
  → Substitution t = aˣ to a Quadratic
• An algebraic root is not a solution until the domain clears it
  → Domain Checks & Counting Solutions
• The argument must be strictly positive — $0$ is not allowed
  → Domain Checks & Counting Solutions
• The GP condition squares the MIDDLE term
  → GP, Chain-Rule & AM-GM Conditions
• Count the 5s, not the 2s — and don't forget 25, 125…
  → Application — Trailing Zeros of a Factorial