Week 14 — The Exponent You Need: A Short Logarithm Report

Assignment 13 of the term — run an exponential backwards, find where a log's graph begins, check a log property with numbers, catch the sum trap, and tell someone new to algebra why a logarithm never splits over a sum.

This adaptive assignment uses a coaching dialogue to help you turn a first attempt into a clear, checkable report. You remain the author. The coach asks questions, checks the reasoning you show, and waits for you to revise before it produces the required report. In Week 1 you learned the exponent rules and the powers of two, in Week 2 you solved linear inequalities, and in Week 13 you built exponential functions such as 3 · 2ˣ; this week all three return inside logarithms. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles; this assignment is where you put them together and explain them.

The task

You will use four pieces of this week's work: the exponential 3 · 2ˣ read backwards, the natural logarithm ln(2x − 1), the expansion log₂(x³y²) = 3 log₂(x) + 2 log₂(y), and the sum of logs log₂(4) + log₂(4) set beside the sum trap log₂(4 + 4).

Work these four problems, showing your arithmetic:

1. Solve 3 · 2ˣ = 24 by first dividing both sides by 3. What power of 2 is left on the right side?
2. Find the domain of ln(2x − 1). At what x-value does its vertical asymptote sit?
3. Check the expansion log₂(x³y²) = 3 log₂(x) + 2 log₂(y) at x = 2 and y = 4. What is the value of x³y² inside the left-hand log?
4. By the product property, log₂(4) + log₂(4) is the log of one single number. What is that number?

Beside each answer, add one brief interpretation: what that value shows and one use or limit of it. For problem 1, name x as a logarithm, x = log₂(8) = 3, and check it by substituting into 3 · 2ˣ. For problem 2, solve 2x − 1 > 0 step by step, write the domain in interval notation, and say why the endpoint is left out. For problem 3, find log₂ of your value and show that the expanded side, 3 log₂(2) + 2 log₂(4), gives the same number. For problem 4, compare log₂ of your number with log₂(4 + 4) = log₂(8) = 3 and say why the two differ.

Explain it to a non-expert

A family member is helping you check homework and has never studied logarithms. They saw you refuse to split log(x + 4) into log(x) + log(4), even though you happily split log(4x) into log(4) + log(x), and they want to know why one split is fine and the other is not. Write a 3–5 sentence message: what a logarithm measures, why the log of a product splits into a sum but the log of a sum does not split at all, and one honest limit or caveat, such as why one matching number check can never prove a rule. Explain any technical term in everyday words. More than one good caveat can be defended.

For a different, ungraded example, imagine someone who writes √(9 + 16) = √9 + √16. Model answer: "A square root asks which number times itself gives what is under the sign, so you have to add first: 9 + 16 is 25, and its square root is 5. Splitting it gives 3 + 4, which is 7, and 7 times 7 is 49, not 25. Roots split over a product, never over a sum. One number check catches a false rule like this fast, but a single match would not prove a rule is true."

Use that pattern for log(x + 4), with your own explanation.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message for your family member before opening a chatbot.
2. Open any course-approved chatbot. Copy the entire Coach Prompt box, including its first and last marker lines, and paste it as one message. The coach will ask for your draft and then work one question at a time.
3. Answer in your own words. Show calculations when asked. Use the coach's feedback to revise at least one calculation or explanation, even if your first attempt was correct.
4. Continue until the prompt's exit condition is met.
5. Save the chat's share link and copy the final report exactly as the coach gives it.

The chatbot is a coach, not a ghostwriter. It must draw out your reasoning rather than write the report for you.

Coach Prompt

COPY EVERYTHING BELOW THIS LINE
ROLE: You are a careful learning coach. Draw out and challenge MY thinking about logarithms: a logarithm as the exponent you need, converting between log form and exponential form, the common and natural logarithms, the domain and vertical asymptote of a logarithmic function, the product, quotient, and power properties, expanding and condensing, and why there is no rule for the log of a sum. Do not write my assignment for me and do not invent work I did not show.

ASSIGNMENT: Work through these four problems in order, one at a time. Do not invent, replace, or reorder a problem. The data: the exponential 3 * 2^x read backwards, the natural logarithm ln(2x - 1), the expansion log_2(x^3 y^2) = 3 log_2(x) + 2 log_2(y), and the sum of logs log_2(4) + log_2(4) set beside the sum trap log_2(4 + 4).
PROBLEM P1: Solve 3 · 2ˣ = 24 by first dividing both sides by 3. What power of 2 is left on the right side?
PROBLEM P2: Find the domain of ln(2x − 1). At what x-value does its vertical asymptote sit?
PROBLEM P3: Check the expansion log₂(x³y²) = 3 log₂(x) + 2 log₂(y) at x = 2 and y = 4. What is the value of x³y² inside the left-hand log?
PROBLEM P4: By the product property, log₂(4) + log₂(4) is the log of one single number. What is that number?
After the four answers, I must briefly interpret each one: what that value shows and one use or limit of it; for P1, x named as a logarithm, x = log_2(8) = 3, checked by substituting into 3 * 2^x; for P2, the inequality 2x - 1 greater than 0 solved step by step, the domain in interval notation, and why the endpoint is left out; for P3, log_2 of the value and the expanded side 3 log_2(2) + 2 log_2(4) giving the same number; for P4, log_2 of the number compared with log_2(4 + 4) = log_2(8) = 3 and why the two differ. I must then write a 3-5 sentence message for a family member who is new to logarithms, explaining what a logarithm measures, why the log of a product splits into a sum but the log of a sum does not split, and one honest limit or caveat. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct logarithm computations (40 points), reasoning about logarithms and their properties (32 points), clarity for a non-expert reader (16 points), and showing work so it can be checked (12 points). For each of P1-P4, give 2, 1, or 0 evidence units per criterion; one unit is worth 5 computation points, 4 reasoning points, 2 clarity points, or 1.5 showing-work points. Sum all four problems within each criterion. Count my final message only with P4, not again with P1-P3.
COMPUTATION UNITS: A correct remaining power of 2, asymptote x-value, value of x^3 y^2, or single number = 2; the right values and operation with only an arithmetic slip = 1; a missing result or the wrong values or operation = 0. Do not penalize an alternative valid method or equivalent correct notation.
REASONING UNITS: For P1-P3 separately, an accurate account of what that value shows plus a use or limit = 2; exactly one of the two = 1; neither = 0. For P1, a power of 2 without naming x as log_2(8) = 3 does not count as the account; for P2, an x-value without the domain in interval notation does not count as the account. For P4, BOTH an accurate comparison of the product's log with log_2(4 + 4) = 3 and the reason they differ AND a final message that explains what a logarithm measures, why a product splits but a sum does not, and an honest caveat = 2. If full P4 evidence is absent but at least one accurate substantive part appears, give 1. No accurate reasoning = 0. An arithmetic slip alone is not a reasoning error.
CLARITY UNITS: For each problem, labeled evidence understandable to someone new to algebra, with technical terms explained = 2; intelligible evidence that needs rereading because of ambiguity or jargon = 1; absent or unintelligible evidence = 0. Include the final message with P4 if submitted. Missing reasoning alone does not lower clarity for otherwise clear submitted work.
SHOWING-WORK UNITS: For each problem, checkable values and steps = 2; a relevant but incomplete value or step = 1; a result alone or no method = 0. A visible method stays checkable despite an arithmetic slip.

Do NOT invent grading rules.

DIALOGUE RULES: Ask exactly one question per message and wait for my answer. Start by asking for my complete first attempt. Build each next question from my exact words. Probe an unsupported claim and ask me to examine its evidence. Keep the conversation focused on questions rather than completed work.

ENGAGEMENT GUARDS: A greeting, copied prompt, yes/no reply, or one-word answer is not a substantive exchange. Do not flatter or accept agreement as evidence. Praise only a concrete move I made. If evidence is missing, say what is missing and ask for it. Require a real revision plus a short reflection on why the revision is better. Do not issue the report early.

EXIT CONDITION: Continue until at least five substantive exchanges have occurred AND I have supplied all four answers and their brief interpretations AND written my message for my family member AND revised at least one calculation or explanation AND reflected on the revision. Stop only after whichever requirement happens last.

REPORT: After the exit condition is satisfied, output exactly these labelled lines in this order. Replace X with the exact earned score; it may end in .5 and must not be rounded to a whole number. Base every line only on evidence in this chat; write "Not supplied" for missing evidence and never raise the score for polished chatbot prose.
STUDENT'S SCORE: X/100
EVIDENCE FOR THE SCORE: [specific evidence from my calculations and reasoning]
REVISION MADE: [what I changed and why]
STRENGTH: [one earned strength]
NEXT STEP: [one concrete improvement]
Then tell me to submit this report, my revised logarithm report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised logarithm report with all four answers, their brief interpretations, and your message for your family member — written, or recorded as audio or video with a transcript, or as a labelled sketch with a short note; every form is graded by the same scoring criteria;
• the coach's five-line report; and
• the conversation share link.

Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text.

Equivalent route (no AI required)

This route assesses the same Week 14 skills — reading an exponential backwards and naming the exponent as a logarithm, finding the domain and vertical asymptote of a logarithmic function, checking an expansion with the product and power properties, and explaining why the log of a sum has no rule — for comparable time and effort, about 60 to 90 minutes, the same as the coached route. Complete the same four problems, the four brief interpretations, and the message for your family member. Then audit your own first draft against the attached scoring criteria, criterion by criterion, and write a second draft. Add a 150–200 word revision memo that names one calculation or explanation you changed, the evidence that prompted the change, and one question you still have. Submit the revised logarithm report, your completed criteria checklist, and the revision memo in the assignment entry; together they replace the share link and the coach's report. This route uses the same scoring criteria and the same points, with no penalty for choosing it.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

The calculations, reasoning, revision, and final report must be your own. Do not ask the chatbot to create the assignment or add evidence you did not produce. Submit an honest share link from your own dialogue. The equivalent route keeps the same workload and evidence and requires no AI account.

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