How many years to reach a target
Compound growth (lesson 5) answers "start with this, grow at this rate - how
much after t years?" The reverse question comes up just as often: "how long
until it reaches a target?" A city planner asking when the population hits the
water system's limit, an investor asking when a portfolio doubles.
The forward formula is target = start * (1 + r)^t. To pull t down out of
the exponent, you need the logarithm - the tool whose whole job is answering
"what exponent produced this?":
t = log(target / start) / log(1 + r)
Read it as: "how many factors of (1+r) fit inside the total growth
(target/start)?" The log function here is the natural logarithm - which one
you use doesn't matter as long as top and bottom use the same one, because
the ratio cancels the choice out.
The answer is almost never a whole number of years - that's fine. 15.9 years
means "during the 16th year."
Your task: a city of 50,000 people grows 3% per year. How many years
until it reaches 80,000? Use log(target/start) / log(1 + r).
You'll practice:
- Using logarithms to solve for an exponent
- Working with a formula where the answer is deliberately not a round number