Week 15 — Solve for the Exponent, Then Check What Survives: A Short Equations Report

Assignment 14 of the term — the last one — match bases, convert a logarithm, time a decaying sample, catch an extraneous candidate, choose the right method for each, and tell someone new to algebra why a correct-looking answer can still fail.

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 2 you learned to check a solution by substitution, in Weeks 11 and 12 you met extraneous solutions, and in Week 14 you learned to convert and condense logarithms; this week all of them return. 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 equation 3^(x + 1) = 27, the logarithmic equation log₂(x − 1) = 3, the decaying sample A = A₀ · e^(−0.15t) with its half-life of about 4.62 hours, and the logarithmic equation log(x) + log(x − 3) = 1.

Work these four problems, showing your arithmetic:

1. Solve 3^(x + 1) = 27 by matching bases. What is x?
2. Solve log₂(x − 1) = 3 by converting it to exponential form. What is x?
3. A sample decays as A = A₀ · e^(−0.15t), with t in hours, so its half-life is ln 2 ⁄ 0.15 ≈ 4.62 hours. How many hours does a 400-gram sample take to fall to 50 grams?
4. Solving log(x) + log(x − 3) = 1 by condensing and converting gives two candidates. What is the candidate that fails the domain check, the extraneous solution?

Beside each answer, add one brief interpretation: what that value shows, the check that confirms it, and the method you chose with one reason a plausible alternative fits less well. For problem 1, rewrite 27 as a power of 3, check your value by substitution, say why setting x + 1 = 27 is wrong, and say why matching bases beats taking a logarithm here. For problem 2, write the domain first, check your value in the original logarithm, and say why setting x − 1 = 3 is wrong. For problem 3, count the halvings, show the multiplication, and say why the starting mass does not change the half-life. For problem 4, name both candidates, check each one in every original argument, and say why the failing one fails.

Choose the method

This is the last assignment of the course, so it asks you to choose before you compute. For each problem, name the move you used — matching bases, converting a logarithm to a power, counting half-lives, or condensing and then running the domain check — and give one reason a plausible alternative fits that problem less well. This is the habit the whole course has built: read the problem, name the move, then check the result.

Explain it to a non-expert

A family member is helping you check homework and has never studied logarithms. They watched you solve log(x) + log(x − 3) = 1, get two numbers, and then cross one out, and they want to know which answer to trust. Write a 3–5 sentence message: why correct algebra can still produce a value that does not work, how the domain check catches it, and one honest limit or caveat, such as why the check must use the original equation and not the condensed one. Explain any technical term in everyday words. More than one good caveat can be defended.

For a different, ungraded example, imagine a friend who solved log₃(x) + log₃(x + 8) = 2 and got x = 1 and x = −9. Model answer: "A logarithm only works on positive numbers, the same way you can't take the square root of a negative number on a regular calculator. When you combine the two logs and solve, the algebra hands you two numbers, but it forgets that rule. Putting −9 back into the original equation would ask for the log of a negative number, which doesn't exist, so −9 is thrown out and 1 is the answer to trust. The catch is that you have to test the original equation, because the combined version hides the problem."

Use that pattern for log(x) + log(x − 3) = 1, with your own explanation.

What comes next

Next comes Week 16, the finals week, with the cumulative final exam and no new material. To prepare, use the final study guide, the Final Practice Exam, and the final exam-prep tutorial in the finals-week module. The choose-the-method habit you practise here is the same habit those materials ask you to use on problems from every week of the course.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations with your method choices, 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 exponential and logarithmic equations: matching bases, isolating a power, taking a logarithm of both sides, converting a logarithmic equation to exponential form, condensing with the product property, the domain check, extraneous solutions, doubling time, and half-life. 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 equation 3^(x + 1) = 27, the logarithmic equation log_2(x - 1) = 3, the decaying sample A = A0 * e^(-0.15t) with t in hours and a half-life of about 4.62 hours, and the logarithmic equation log(x) + log(x - 3) = 1, where log means base 10.
PROBLEM P1: Solve 3^(x + 1) = 27 by matching bases. What is x?
PROBLEM P2: Solve log₂(x − 1) = 3 by converting it to exponential form. What is x?
PROBLEM P3: A sample decays as A = A₀ · e^(−0.15t), with t in hours, so its half-life is ln 2 ⁄ 0.15 ≈ 4.62 hours. How many hours does a 400-gram sample take to fall to 50 grams?
PROBLEM P4: Solving log(x) + log(x − 3) = 1 by condensing and converting gives two candidates. What is the candidate that fails the domain check, the extraneous solution?
After the four answers, I must briefly interpret each one: what that value shows, the check that confirms it, and the method I chose with one reason a plausible alternative fits less well; for P1, 27 rewritten as a power of 3, a substitution check, why setting x + 1 = 27 is wrong, and why matching bases beats taking a logarithm here; for P2, the domain first, a check in the original logarithm, and why setting x - 1 = 3 is wrong; for P3, the halvings counted, the multiplication shown, and why the starting mass does not change the half-life; for P4, both candidates, each checked in every original argument, and why the failing one fails. I must then write a 3-5 sentence message for a family member who is new to logarithms, explaining why correct algebra can still produce a value that does not work, how the domain check catches it, and one honest limit or caveat. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct exponential and logarithmic computations (40 points), reasoning about methods, checks, and candidates (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 value of x, number of hours, or failing candidate = 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 with its check, plus the method chosen and why one alternative fits less well = 2; exactly one of the two = 1; neither = 0. For P1, a value without the substitution check does not count as the account; for P2, a value without the domain or the conversion does not count as the account. For P4, BOTH an accurate check of both candidates with the reason the failing one fails AND a final message that explains how correct algebra can produce a value that does not work, how the domain check catches it, 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 equations report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised equations report with all four answers, their brief interpretations and method choices, 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 15 skills — solving an exponential equation by matching bases, solving a logarithmic equation by converting it, finding a decay time from a half-life, rejecting an extraneous solution with the domain check, and choosing the method that fits each problem — 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 with your method choices, 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 equations 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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