Exponential growth and logarithms: the same factor over equal time

Where we are on the map

Functions now move from a constant difference to a constant factor.

Begin with a familiar image

If bacteria double each hour, the added amount is not constant, but the multiplication factor is.

Precise meaning

In N=N₀aᵗ, N₀ is the initial value and a is the factor per step. logₐb asks for the power of a that equals b.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

initial value → time step → constant factor → power → logarithm finds time

Worked example

N=500×2ᵗ; at t=3, N=4000. In reverse, 2³=8, so log₂8=3.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

After three doublings, writing 500+3×2=506 is wrong: every step doubles the entire previous value.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. A quantity starts at 1000 and grows by a factor of 1.5 every 4 hours. Find its value after 12 hours and when it reaches 3375.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

Sequences list step-by-step values and connect linear and exponential patterns.

Course contents