Week 15 — Solve for the Exponent, Then Check What Survives: A Short Equations Report — Traditional alternative (no AI)

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

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 week you learned to solve an exponential equation by matching bases or by taking a logarithm of both sides, to solve a logarithmic equation by converting it or by condensing it first, to reject an extraneous solution with the domain check, and to find doubling times, tripling times, and half-lives. 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. In this assignment you work four short problems, interpret each one, choose and defend a method for each, and explain to someone who has never studied logarithms why a correct-looking answer can still fail. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles.

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.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

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.

Deliverables

Submit a single equations report containing:

• the four answers, each with the arithmetic shown, a brief interpretation of what that value shows and its check, and your method choice with one reason an alternative fits less well;
• your message for your family member, explaining how correct algebra can produce a value that does not work, how the domain check catches it, and one honest caveat; and
• a one-line restatement of the equations and model you used, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes with a transcript, or as a labelled sketch with a short note. Every form is graded by the same scoring criteria. 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.

How you'll be graded

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

Academic integrity and AI use

This version is designed to be completed without AI: do the computations, the interpretations, and the writing yourself, without a chatbot or other AI tool. You may use a calculator or a free graphing calculator such as Desmos to check your arithmetic and your answers. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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