Exponential change and logarithms
Model repeated percentage change and solve for the number of periods.
Before you begin
- Index laws
- Percentage multipliers
Model equal multiplicative changes
A quantity changing by the same percentage each period follows an exponential model A = A₀bᵗ. Here A₀ is the initial amount and b is the per-period multiplier. Equal additive changes give a linear model instead. The time unit must match the multiplier: a yearly growth rate cannot be used with a time measured in months without adjustment.
Interpret logarithms as exponents
log_b(a) is the exponent needed to make bᵗ = a, with b > 0, b ≠ 1 and a > 0 in real-number work. To solve A₀bᵗ = A, divide by A₀ first and then use t = log(A/A₀)/log(b). A model's calculated time may need rounding upward if only complete periods are possible.
Worked example
A model population starts at 200 and doubles every period. After how many periods does it reach 1,600?
Show the worked solution
- Write 200 × 2ᵗ = 1,600.
- Divide by 200 to obtain 2ᵗ = 8.
- Since 8 = 2³, the required exponent is 3.
3 periods.
Try it yourself
Which multiplier represents a 6% decrease each year?
Common mistakes
- Repeated percentage change is multiplicative, not a fixed subtraction.
What can you explain now?
A quantity of 500 decreases by 20% per period. Find it after two periods.
Compare with the explanation
320.
Use 500 × 0.8² = 320. The second decrease acts on 400, not on the initial 500.
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Common Core mathematics standards ↗