maths1-week6

Week map

Logarithm notation

• $\log_a x$ → “log base $a$ of $x$” → the exponent $y$ such that $a^y = x$ → requires $a > 0$, $a \neq 1$, $x > 0$.
• $\ln x$ → natural log → base $e$ → $\ln x = \log_e x$.
• $\log_{10} x$ or $\log x$ → common log (context-dependent).

Key identities

• $a^{\log_a x} = x$ for $x > 0$.
• $\log_a(a^x) = x$ for all real $x$.
• $\log_a 1 = 0$; $\log_a a = 1$.

Log laws

• Product: $\log_a(xy) = \log_a x + \log_a y$.
• Quotient: $\log_a(x/y) = \log_a x - \log_a y$.
• Power: $\log_a(x^n) = n \log_a x$.
• Change of base: $\log_a x = \frac{\log_b x}{\log_b a}$ for any valid base $b$.

Graph features

• Domain $(0, \infty)$; vertical asymptote at $x = 0$.
• Passes through $(1, 0)$ and $(a, 1)$.
• Reflection of $y = a^x$ over $y = x$.

Pattern families

Easy — Evaluate simple logs

Medium — Apply log laws

Hard — Solve log and exponential equations

Worked mini-examples

Traps

• Taking $\log$ of negative number or zero — undefined.
• Using $\log(x + y) = \log x + \log y$ — false; product law only.
• Forgetting $0!$-style edge: $\log_a 1 = 0$, not undefined.
• Dividing logs incorrectly: $\log x / \log y \neq \log(x/y)$.
• Accepting solution that makes log argument $\leq 0$.

Diagnostic (try yourself)

1. Evaluate $\log_5 125$ and $\log_3(1/9)$.
2. Expand $\log_2(8x^3)$ using log laws.
3. Use change of base to express $\log_8 4$ as a fraction.
4. Solve $\log_{10}(x) = 2$. State the domain condition.
5. True or false: $\ln(e^5) = 5$? What is $\ln 1$?