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Teaching video
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Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Preserve each logarithm's positivity requirement when solving.
- Justify the conclusion "x=3 is the only solution" using the stated assumptions.
Before you start
Logarithm laws and quadratics.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Solve ln(x−1)+ln(x+1)=ln 8.
Why this math matters
Preserve each logarithm's positivity requirement when solving. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- ln is the real natural logarithm.
- The product rule is used only for positive factors.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Combine logarithms without admitting forbidden inputs
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Solve ln(x−1)+ln(x+1)=ln 8.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Both inputs must be positive, so x>1
The individual logarithms impose more than positivity of their product.
Work through the mathematics
ln((x−1)(x+1))=ln8 gives x²−1=8
The product law is valid on the retained domain.
Check the conclusion
x=3 is the only solution
The algebraic candidate −3 violates x>1, whereas ln2+ln4=ln8 checks three.
The result
x=3 is the only solution
The algebraic candidate −3 violates x>1, whereas ln2+ln4=ln8 checks three.
Common mistakes to catch
- A combined logarithm can hide original domain exclusions.
- Logarithms of a sum do not split into sums of logarithms.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Why does x=−3 make the product positive but fail the original equation?
Show a hint
Inspect each logarithm separately.
Reveal answer and explanation
Both original inputs are negative
A positive product does not make the two real logarithms defined.
Practice 2
Solve ln x=0.
Show a hint
Rewrite as x=exp(0).
Reveal answer and explanation
One
The logarithm's input remains positive.
Take the idea with you
Check every original input condition after compressing a formula.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Recover an exponential model from two readings
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