Week 6 — logarithmic functions

Quiz 2 scope: Weeks 1–8 per IITM May 2026 foundation courses. Source baseline: IITM BS admissions important-dates calendar · May 2026 cycle. Times on assessments are operational conventions — verify hall ticket.

Week map

Classify → Represent → Execute → Trap-check

• Recognize: Ask: What does $\log_2 8$ ask in words?
• Procedure: Rewrite as exponent question: "what power of base gives this number?" Use $\log_a 1 = 0$, $\log_a a = 1$. For base 10 or $e$, connect to familiar values.
• Variations / traps: Watch for: Using $\log(x+y) = \log x + \log y$ (false).

Formula chain (compressed)

log = inverse of exp → log laws → change of base → solve log equations.

1. Definition — y=log_a(x) ⇔ a^y=x — a>0, a≠1, x>0
2. Product — log(ab)=log a + log b — same base
3. Quotient — log(a/b)=log a − log b — same base
4. Power — log(a^k)=k log a — pull exponent out
5. Change of base — log_a b = log_c b / log_c a — calculator base

Deep study

Mathematics I · Week 6 — Logarithms

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 $\ln 1$?

ChatGPT prep archive

• Chatty reference archive

Archived import for extra depth — complements the notes above, not official IITM material.

Core concepts

• Definition: $y = \log_a x$ means $a^y = x$, with $a>0$, $a\neq 1$, $x>0$.
• Common laws: $\log_a(xy)=\log_a x + \log_a y$; $\log_a(x^n)=n\log_a x$; change of base.
• Graph: reflection of exponential over $y=x$; vertical asymptote $x=0$.
• Solving: isolate log or exponent; check solutions in original domain.

Notation & vocabulary

Pattern families

Easy — Evaluate simple logs

Medium — Expand or condense

Hard — Log equation with domain

Drill these on the pattern atlas — filter to week 6.

Traps

• Using $\log(x+y) = \log x + \log y$ (false).
• Forgetting $x>0$ in $\log x$.
• Exponentiating without applying to entire side.
• Change-of-base with inverted fraction.

Retrieval prompts

1. What does $\log_2 8$ ask in words?
2. Why must log arguments be positive?
3. How do you solve $2^x = 10$ using logs?

Practice loop

1. Read Deep study (if present) or core concepts once.
2. Recite the formula chain without looking.
3. Open one easy pattern on the interactive atlas for week 6.
4. Attempt without solutions; mark studied after an honest try.
5. Say one trap aloud before closing the tab.

Related

• Full weeks 1–8 prep guide
• Pattern atlas note
• Formula chain desk