LEMMA

Exponentials & logs

Exponential growth and decay

ALEKS placement

Growth multiplies by a constant factor each period — not by a constant amount.

What this covers

A=A0​bt/p,A=A0​ekt
  • Identify the initial amount A₀.
  • Find the growth factor: 1 + rate for growth, 1 - rate for decay.
  • Put the exponent in units of the compounding period.
  • Evaluate the power last, after the factor and exponent are right.

Worked example

Worked example

A population of 500 grows 4% per year. Find it after 10 years.

  1. A=500(1.04)10The growth factor is 1 + 0.04.
  2. A≈7401.04¹⁰ ≈ 1.48, and 500 times that is about 740.
Another worked example

A 500 mg sample loses 20% each hour. Model its amount after t hours and find its amount after two hours.

  1. b=1−0.20=0.80The factor is the fraction remaining, not the fraction lost.
  2. A(t)=500(0.8)t,t≥0The initial amount is 500 mg, and t counts hours to match the hourly rate.
  3. A(2)=500(0.8)2=320 mgMultiply by 0.8 twice. Subtracting 100 mg twice would wrongly treat the change as additive.

A common mistake

A mistake Lemma catches

From

A=500(0.04)10A=500(1.04)10

“4% growth keeps the 100% you had and adds 4%, so each year multiplies by 1 + 0.04 = 1.04: after one year, 500 · 1.04 = 520. Multiplying by 0.04 keeps only 4%: 500 · 0.04 = 20 after one year — a collapse, not growth.”

Try one

Sample problem

$100 grows by 10% per period for 2 periods. Find the final amount.

A=100(1+101​)2

Practice growth & decay free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

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