Question

the population is 300,000 with a 3% annual growth rate how long until the population will reach 1.2 million?

Answer 1

?? \(300000(1.03)^{t} = 1200000\) Use your best logarithms to solve for 't'.

It's a WHOLE LOT less than 100 years.

Answer 2

After 1 year, population = 300,000 x 1.03
after 2 years, population = 300,000 x 1.03 x 1.03
= 300,000 x (1.03)^2
after 3 years, population = 300,000 x 1.03 x 1.03 x 1.03
= 300,000 x (1.03)^3

Note the pattern forming.
After n years, population = 300,000 x (1.03)^n
(^ is to the power of)

Problem now written as:
300,000 x (1.03)^n = 1,200,000
divide both sides of equation by 300,000
1.03^n = 4
take ln of both sides (ln button calculator)
ln(1.03^n) = ln(4)
now use logarithm rule lna(b^n) = n x lna(b)
This rule can be proven for example with log10(10^2) = 2 x log10(10). Log10 is log button on calc.

So n x ln (1.03) = ln(4)
divide both sides by ln (1.03)
n = ln(4) divided by ln(1.03)
calculator does this. Answer = 46.8995445 years. So after 47 years of annual growth rate of 3%, population reaches 1.2 million.

A check of this answer is to sub solution into the original problem.
300,000 x (1.03)^46.8995445 = 1.2 million

Note about using logarithms to solve.
I always find it hard to remember what way to write these problems so I type log10 (1000) in the calculator to work out the form in which to write the problem.
Note log10(1000) = 3. This means 10^3 = 1000
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