Learn with Amar
Teaching video
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Browse grades and teaching videos01 · Read and understand
What you will learn
- Isolate an exponential factor.
- Interpret a logarithm as an exponent.
- Apply a whole-cycle restriction to a threshold.
Before you start
Evaluate powers, solve equations by division, and use a calculator's logarithm function.
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
A fictional simulation starts with 40 tokens and doubles the count every two days. After how many recorded doubling cycles does it reach 640 tokens, and how many days is that?
Why this math matters
Exponential rules answer 'how much after this many cycles?' Logarithms reverse the question: 'how many cycles produce this amount?' A model's observation schedule matters when the algebra returns a fraction of a cycle.
Set up the model
A useful answer starts with clear assumptions:
- The simulation doubles exactly every two days with no losses.
- Recorded counts occur only after complete cycles; this is not a prediction of real population growth.
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.
Use a logarithm to count doubling cycles
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A fictional simulation starts with 40 tokens and doubles the count every two days. After how many recorded doubling cycles does it reach 640 tokens, and how many days is that?
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.
Write and simplify the model
40 × 2ⁿ = 640; 2ⁿ = 16
n is the number of complete doubling cycles. Dividing by the initial count isolates the growth factor.
Recover the exponent
n = log₂(16) = 4
The logarithm asks which power of two equals sixteen. A calculator can also use ln(16)/ln(2), with the same result.
Convert cycles to days
4 cycles × 2 days/cycle = 8 days
The token sequence is 40, 80, 160, 320, 640. Counting four transitions confirms the exponent and the elapsed time.
The result
The simulation reaches 640 tokens after 4 cycles, or 8 days.
For a target of at least 300, log₂(300/40) is about 2.907 cycles. The first recorded count meeting that threshold is cycle three, not an invented partial cycle.
Common mistakes to catch
- Divide by the starting amount before taking the logarithm of the growth factor.
- Do not confuse the number of cycles with days when one cycle lasts two days.
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
When does the same simulation first reach 320 tokens?
Show a hint
Find the exponent in 2ⁿ = 320/40.
Reveal answer and explanation
3 cycles, or 6 days
320/40 = 8 = 2³, so three doublings are needed.
Practice 2
What is the first recorded time with at least 300 tokens?
Show a hint
Compare the complete-cycle counts around 300.
Reveal answer and explanation
6 days
At four days the count is 160; at six days it is 320. The first recorded count above the target is therefore at six days.
Take the idea with you
After solving an exponential equation, check whether its exponent measures continuous time, whole cycles, or another restricted input.
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 rectangle dimensions by factoring
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