Learn with Amar
Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Use logarithms to invert an exponential transformation.
- Justify the conclusion "f⁻¹(y)=log₃((y−5)/2), with y>5" using the stated assumptions.
Before you start
Exponential functions and logarithms.
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
Find the inverse of f(x)=5+2·3ˣ.
Why this math matters
Use logarithms to invert an exponential transformation. 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:
- The original domain is all real numbers.
- The base three is positive and not one.
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.
Undo a shifted exponential and state the recovered domain
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find the inverse of f(x)=5+2·3ˣ.
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
y−5=2·3ˣ
Remove the vertical shift before undoing the scaling.
Work through the mathematics
(y−5)/2=3ˣ ⇒ x=log₃((y−5)/2)
The logarithm reverses exponentiation only for a positive input.
Check the conclusion
f⁻¹(y)=log₃((y−5)/2), with y>5
The inverse's input range is the original exponential's shifted range.
The result
f⁻¹(y)=log₃((y−5)/2), with y>5
The inverse's input range is the original exponential's shifted range.
Common mistakes to catch
- Subtract the shift before dividing by the scale.
- An asymptotic output is not necessarily an attained output.
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
Find f⁻¹(11).
Show a hint
Compute (11−5)/2.
Reveal answer and explanation
One
log₃3=1.
Practice 2
Can the inverse take y=5?
Show a hint
Its logarithm input would be zero.
Reveal answer and explanation
No
f approaches five but never attains it for a finite real x.
Take the idea with you
Recover a model input from a positive sensor reading above its baseline.
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: Control the tail of an infinite geometric sum
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