Week 14 — The Exponent You Need: A Short Logarithm Report — Traditional alternative (no AI)

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

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 week you learned to read a logarithm as the exponent you need, to convert between log form and exponential form, to use the common and natural logarithms, to find the domain and vertical asymptote of a logarithmic function, and to expand and condense with the product, quotient, and power properties. 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. In this assignment you work four short problems, interpret each one, and explain to someone who has never studied logarithms why a log never splits over a sum. 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 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.

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.

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.

Deliverables

Submit a single logarithm report containing:

• the four answers, each with the arithmetic shown and a brief interpretation of what that value shows and one use or limit of it;
• your message for your family member, explaining what a logarithm measures, why the log of a product splits but the log of a sum does not, and one honest caveat; and
• a one-line restatement of the expressions and equations 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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