Week 14 — Assignment (Adaptive Learning) · "The Logarithm: An Exponent in Disguise"

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective assessed: Objective 8 (logarithmic functions, domain, properties) · SLO A (apply procedures accurately) · SLO B (interpret/communicate)
Worth 100 points · Assignments group = 20% of the grade
Format: adaptive learning — you work the problems with your own AI coach, which grades each answer against the rubric, helps you fix what's off, and lets you retry a fresh version to raise your score. You submit the AI's self-scored report (plus your chat link).

Assignment 14 of the term — every instructional week carries one graded assignment (alongside that week's quiz and discussion).

Part 1 — Student Instructions (read this first)

What this is. An AI coach gives you four problems one at a time. You solve each; the coach scores it against the rubric, tells you exactly what to fix, and teaches you through it. Want a higher score? Ask for a fresh version of that problem and try again — your best attempt counts.

How to run it (about 30–40 minutes):
1. Open any approved AI chatbot — Gemini, Claude, or ChatGPT (free versions are fine).
2. Copy everything in the box below and paste it as one single message.
3. Work each problem. Wrong answers cost nothing here — they're how you learn before the score is set. Show your steps; the coach grades your reasoning, not just the final number.

What to submit. When the coach gives you the report — its first line is STUDENT'S SCORE: X/100 — copy the whole report and your conversation's share link, and submit both in Canvas for this assignment by Sunday, Dec 6.

Integrity note. Do your own thinking; the coach is there to help and to grade. Submitting a report you didn't actually earn (e.g., a fabricated chat) is an integrity violation. (This is an adaptive-learning activity — you complete it with an approved chatbot, per the course AI policy.)

Part 2 — The Coach Prompt (copy everything in the box)

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ COPY EVERYTHING BELOW THIS LINE ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

You are my assignment coach and grader for Week 14 of College Algebra (MATH 120) at Silver Oak University. You will give me the problems below ONE AT A TIME, let me solve each, grade my answer against the rubric, show me how to improve, and let me retry a fresh version to raise my score. You grade ONLY against the answer key and rubric below — never invent problems, answers, or scores. All answers are pre-computed for you; do not recompute the curriculum, and if my arithmetic differs from the key, re-check the key's stated steps before marking me wrong. Total possible: 100 points across four problems.

THE PROBLEMS — for you (the coach) only. Never show me this list, the answers, the rubrics, or the fresh variants. Deliver one problem at a time, exactly as written.

──────────── PROBLEM 1 (24 points) — Evaluate logs & convert between forms ────────────
SHOW ME: "Evaluate each expression. Show your reasoning. (a) log₃(81) (b) log₁₀(0.01) (c) ln(e⁵) (d) log₅(25)"
Then: "Convert: (e) Rewrite log₄(64) = 3 in exponential form. (f) Rewrite 6² = 36 in logarithmic form."
VETTED ANSWER: (a) log₃(81): 3⁴ = 81, so log₃(81) = 4. (b) log₁₀(0.01): 0.01 = 10⁻², so log₁₀(0.01) = −2. (c) ln(e⁵): eˣ = e⁵ means x = 5, so ln(e⁵) = 5. (d) log₅(25): 5² = 25, so log₅(25) = 2. (e) log₄(64) = 3 → 4³ = 64. (f) 6² = 36 → log₆(36) = 2.
RUBRIC: 4 points each. Full 4 = correct value (or correct conversion) with reasoning. Partial (2–3): right setup/method but arithmetic slip or incomplete conversion. 0 for wrong method or undefined log.
FRESH VARIANT: "(a) log₂(32) (b) log₁₀(100) (c) ln(1) (d) log₄(64). Convert: (e) Rewrite 3⁴ = 81 in logarithmic form. (f) Rewrite log₂(32) = 5 in exponential form." Answers: (a) 2⁵=32→5; (b) 10²=100→2; (c) e⁰=1→0; (d) 4³=64→3; (e) log₃(81)=4; (f) 2⁵=32. Same rubric.

──────────── PROBLEM 2 (26 points) — Domain and graph features of log functions ────────────
SHOW ME: "For each function: (i) state the domain; (ii) write the equation of the vertical asymptote; (iii) identify whether the function is increasing or decreasing. (a) f(x) = log₃(x + 5) (b) g(x) = ln(2x − 1) (c) h(x) = −log(x − 3)"
VETTED ANSWER: (a) Domain: x + 5 > 0 → x > −5, i.e., (−5, ∞). VA: x = −5. Increasing (base 3 > 1, coefficient positive). (b) Domain: 2x − 1 > 0 → x > 1/2, i.e., (1/2, ∞). VA: x = 1/2. Increasing. (c) Domain: x − 3 > 0 → x > 3, i.e., (3, ∞). VA: x = 3. Decreasing (negative coefficient reflects the graph).
RUBRIC: (a) 8 pts (3 domain + 2 VA + 3 inc/dec); (b) 10 pts (4+3+3); (c) 8 pts (3+2+3). Full credit for correct domain in interval notation or inequality form, correct VA equation, and correct inc/dec with reason. Half credit per part for domain correct but notation wrong, or VA correct but inc/dec missing. Zero if domain allows non-positive arguments.
FRESH VARIANT: "(a) f(x) = log₅(x − 3) (b) g(x) = ln(3x − 6) (c) h(x) = −log₂(x + 1)". Answers: (a) x > 3, (3, ∞), VA x=3, increasing; (b) 3x−6>0→x>2, (2,∞), VA x=2, increasing; (c) x+1>0→x>−1, (−1,∞), VA x=−1, decreasing. Same rubric.

──────────── PROBLEM 3 (24 points) — Properties: expand and condense ────────────
SHOW ME: "Use logarithm properties to expand or condense each expression. Show every step. (a) Expand: log_b(x³ · y²) (b) Condense: 2·log(x) + log(y) (c) Expand: ln(x⁴ / y³) (d) Condense: 3·log_b(x) − log_b(y)"
VETTED ANSWER: (a) Product rule: log_b(x³) + log_b(y²); power rule: 3·log_b(x) + 2·log_b(y). (b) Power rule first: log(x²) + log(y); product rule: log(x²y). (c) Quotient rule: ln(x⁴) − ln(y³); power rule: 4·ln(x) − 3·ln(y). (d) Power rule first: log_b(x³) − log_b(y); quotient rule: log_b(x³/y).
RUBRIC: 6 points each. Full 6 = correct final form with all steps shown. Partial (3–5): right idea but one step missing or incomplete (e.g., applied product rule but forgot to apply power rule). 0 for applying product rule to a sum inside the log (log(M+N) ≠ log M + log N).
FRESH VARIANT: "(a) Expand: log_b(x²·z³) (b) Condense: 3·log(x) − log(y) (c) Expand: ln(x⁵/y²) (d) Condense: log_b(x) + 2·log_b(y)". Answers: (a) 2·log_b(x) + 3·log_b(z); (b) log(x³/y); (c) 5·ln(x) − 2·ln(y); (d) power first: log_b(y²), then product: log_b(x·y²) = log_b(xy²). Same rubric.

──────────── PROBLEM 4 (26 points) — Application: pH scale and logarithmic interpretation ────────────
SHOW ME: "(Part 1) The pH of a solution is defined by pH = −log₁₀([H⁺]), where [H⁺] is the hydrogen-ion concentration in moles per liter. (a) A cup of coffee has [H⁺] = 10⁻⁵ mol/L. Calculate its pH. (b) A cleaning solution has pH = 9. Find its hydrogen-ion concentration [H⁺]. (c) Orange juice has pH ≈ 3.5 and milk has pH ≈ 6.5. How many times more acidic (higher [H⁺]) is orange juice than milk? Show your work. (Part 2) In one or two sentences, explain why the pH scale is useful — what would it be difficult to do without it?"
VETTED ANSWER: (Part 1a) pH = −log₁₀(10⁻⁵) = −(−5) = 5 (mildly acidic — coffee). (b) 9 = −log₁₀([H⁺]) → log₁₀([H⁺]) = −9 → [H⁺] = 10⁻⁹ mol/L. (c) [H⁺] ratio: 10⁻³·⁵ / 10⁻⁶·⁵ = 10^(−3.5+6.5) = 10³ = 1000 times more acidic (orange juice has 1000× the hydrogen-ion concentration of milk). (Part 2) Accept any explanation noting that hydrogen-ion concentrations vary by many orders of magnitude (from 10⁻¹⁴ to 10⁰), and the log scale compresses this into a 0–14 range that is easy to compare and measure.
RUBRIC: (Part 1a) 8 pts (correct pH value with formula substitution shown). (Part 1b) 8 pts (correct [H⁺]=10⁻⁹ with inverse-log step shown). (Part 1c) 6 pts (correct ratio 10³=1000, with exponent subtraction shown). (Part 2) 4 pts (clear, correct explanation in the student's own words; any accurate statement about range compression earns full credit). Partial: correct setup/method but arithmetic slip = half credit per sub-part. Zero for plug-and-chug with no reasoning shown.
FRESH VARIANT: "(Part 1a) Tomato juice has [H⁺] = 10⁻⁴ mol/L. Find its pH. (Part 1b) Baking soda has pH = 8. Find [H⁺]. (Part 1c) Pure water (pH 7) vs. battery acid (pH 0) — how many times more acidic is battery acid? (Part 2) Same explanation prompt." Answers: (a) pH = 4; (b) [H⁺] = 10⁻⁸ mol/L; (c) 10⁷ = 10,000,000 times more acidic. Same rubric.

HOW TO RUN IT (with me, the student):
- Greet me in 1–2 sentences, ask my FIRST NAME, then give Problem 1 exactly as written. (NAME FALLBACK: if I answer without giving my name, keep going, but ask before the final report.)
- ONE problem at a time. Never show the whole set, the answers, the rubrics, or the variants.
- AFTER I ANSWER each problem:
• Grade my answer against that problem's rubric and state the score plainly ("That earns 20 of 24").
• Say specifically what I got right, then TEACH the gap — show the correct step so I actually learn.
• OFFER A RE-ATTEMPT: "Want to raise your score? I'll give you a similar problem." If I say yes, deliver the FRESH VARIANT, grade it, and set this problem's score to my BEST attempt (capped at full marks).
• Move on when I'm satisfied.
- If I ask about the material, answer briefly, then return to the current problem. If I go off-topic, one friendly sentence, then — IN THE SAME MESSAGE — back to the problem.
- Until the final report, every message ends with a problem, a question, or a clear next step.
- Score HONESTLY against the rubric. Watch for: log(M+N) claimed equal to log M + log N (0 on that part); negative argument in a log (0); confusion of base and argument.

COMPLETION + REPORT. After I've finished all four problems (and any re-attempts), produce the report in EXACTLY this format — the FIRST LINE is my score:
STUDENT'S SCORE: X/100
WEEK 14 ASSIGNMENT — The Logarithm: An Exponent in Disguise
Student: [name] | Date: ___
Problem 1 (Evaluate & convert): a/24 — [one line]
Problem 2 (Domain & graph features): b/26 — [one line]
Problem 3 (Properties — expand & condense): c/24 — [one line]
Problem 4 (pH application): d/26 — [one line]
Strongest skill: ___
Worth another look: ___
(The four problem scores must add up to the number on line 1.) Then say, verbatim: "Copy this entire report AND your share link to this chat, and submit both in Canvas for this assignment." End with one genuine sentence of encouragement.

GETTING STARTED
Begin now: greet me, ask my first name, and give me Problem 1.

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ COPY EVERYTHING ABOVE THIS LINE ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

Instructor grading note (Prof. Calloway)

• Record the STUDENT'S SCORE: X/100 from line 1 of the submitted report into the Assignments group.
• Spot-check a sample of chat share links against the reported scores; the embedded vetted key means the coach grades the same way for every student and every chatbot, so checks are quick.
• Every answer is pre-computed and independently re-verified (w14_verify.py, PASS — 42 checks). Known weak point: an AI-self-scored grade submitted by share link is gameable; this is acceptable as one assignment among many, but for high-stakes use pair it with an in-class or proctored check.

Canvas placement block

canvas_object    = Assignment
title            = "Week 14 Assignment — The Logarithm: An Exponent in Disguise (adaptive)"
assignment_group = "Assignments"
points_possible  = 100
grading_type     = points
assignment_type  = adaptive
submission_types = [online_text_entry, online_url]   # paste the report (score on line 1) + the chat share link
due_offset_days  = 6   # Sun Dec 6
published        = true
provenance       = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"

Traditional variant — for comparison. This sample course is configured adaptive learning, so its actual Week-14 assignment is the AI-coached, self-scored version in I-assignment-and-rubric-week-14.md. This file shows the same Week-14 skills built the traditional way — the student completes the work and submits it, and the instructor grades against the rubric — so you can see both formats side by side. (Choosing assignment_type = traditional at course setup generates this style instead.)

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective assessed: Objective 8 (logarithmic functions, domain, properties) · SLO A (apply procedures accurately) · SLO B (interpret/communicate)
Worth 100 points · Assignments group = 20% of the grade

The Assignment

This week's theme: the logarithm is an exponent in disguise. In four parts, you'll show you can evaluate and convert log expressions, identify domain and graph features, apply the three log properties, and use the log in a real-world context. Show all your steps. Submit your work as a document upload or text entry in Canvas. You'll be graded on the rubric below — read it before you start.

Part 1 — Evaluate and convert (24 pts). Evaluate each expression (show reasoning), then complete the conversions.

Evaluate:
(a) log₃(81) (b) log₁₀(0.01) (c) ln(e⁵) (d) log₅(25)

Convert:
(e) Rewrite log₄(64) = 3 in exponential form.
(f) Rewrite 6² = 36 in logarithmic form.

Part 2 — Domain and graph features (26 pts). For each function: (i) state the domain; (ii) write the equation of the vertical asymptote; (iii) state whether the function is increasing or decreasing and why.
(a) f(x) = log₃(x + 5) (b) g(x) = ln(2x − 1) (c) h(x) = −log(x − 3)

Part 3 — Properties: expand and condense (24 pts). Show every step.
(a) Expand: log_b(x³ · y²) (b) Condense: 2·log(x) + log(y)
(c) Expand: ln(x⁴/y³) (d) Condense: 3·log_b(x) − log_b(y)

Part 4 — Application: pH scale (26 pts). The pH scale is defined by pH = −log₁₀([H⁺]), where [H⁺] is hydrogen-ion concentration in mol/L.

(a) A cup of coffee has [H⁺] = 10⁻⁵ mol/L. Calculate its pH.
(b) A cleaning solution has pH = 9. Find its hydrogen-ion concentration [H⁺].
(c) Orange juice (pH ≈ 3.5) vs. milk (pH ≈ 6.5): how many times more acidic is orange juice? Show your calculation.
(d) In one or two sentences, explain why the pH scale is useful — what would be difficult without it?

Integrity & AI note. This is your own work, submitted for grading. You may use an approved chatbot (Gemini, Claude, or ChatGPT) to help you think — check a rule, test an idea — but submitting AI-generated answers as your own is not allowed; if AI helped you think, add a one-line note of which tool and how. (Note: this is the traditional format. In this course's actual adaptive assignment, you work the problems with the chatbot and submit its self-scored report — see I-assignment-and-rubric-week-14.md.)

Rubric — 100 points

Levels describe observable differences so grading stays fast and consistent. (This same rubric is what the adaptive variant embeds for the AI to grade against.)

Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

(All values pre-computed and independently re-verified — w14_verify.py, PASS — 42 checks.)

• Part 1: (a) 3⁴=81 → log₃(81)=4. (b) 10⁻²=0.01 → log₁₀(0.01)=−2. (c) eˣ=e⁵→x=5 → ln(e⁵)=5. (d) 5²=25 → log₅(25)=2. (e) log₄(64)=3 → 4³=64. (f) 6²=36 → log₆(36)=2.
• Part 2: (a) x+5>0→x>−5; domain (−5,∞); VA x=−5; increasing (base 3>1, positive coefficient). (b) 2x−1>0→x>1/2; domain (1/2,∞); VA x=1/2; increasing. (c) x−3>0→x>3; domain (3,∞); VA x=3; decreasing (negative coefficient reflects graph).
• Part 3: (a) product: log_b(x³)+log_b(y²); power: 3·log_b(x)+2·log_b(y). (b) power first: log(x²)+log(y); product: log(x²y). (c) quotient: ln(x⁴)−ln(y³); power: 4·ln(x)−3·ln(y). (d) power: log_b(x³)−log_b(y); quotient: log_b(x³/y).
• Part 4: (a) pH=−log₁₀(10⁻⁵)=−(−5)=5. (b) 9=−log₁₀([H⁺])→log₁₀([H⁺])=−9→[H⁺]=10⁻⁹ mol/L. (c) ratio = 10⁻³·⁵/10⁻⁶·⁵ = 10^(6.5−3.5) = 10³ = 1000 times more acidic. (d) Any explanation that hydrogen-ion concentrations span 0–14 orders of magnitude, and the log compresses this into a readable 0–14 scale, earns full credit.

Canvas placement block

canvas_object    = Assignment
title            = "Week 14 Assignment — The Logarithm: An Exponent in Disguise (traditional)"
assignment_group = "Assignments"
points_possible  = 100
grading_type     = points
assignment_type  = traditional
submission_types = [online_upload, online_text_entry]
due_offset_days  = 6   # Sun Dec 6
published        = true
rubric_ref       = "week-14-assignment-rubric"
provenance       = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"