Applications of Indices Surds and Logarithms

Question

Given that the world population can be estimated by using the formula P = 6.9(1.010)^t, where t is the number of years after 2019 and P is the world population in billions of people.

i) Convert the given formula to an equation in the form of logarithm.

ii) Estimate

a) the population in the year 2030,

b) by what year will the population be double of that in 2019.

Answer 1

i) Convert the given formula to an equation in the form of logarithm

P = 6.9(1.010)^t

P/6.9 = (1.010)^t

log_1.010(P/6.9) = t

ii) Estimate a) the population in the year 2030,

P = 6.9(1.010)^t

P = 6.9(1.010)^{(2030 - 2019)}

P = 6.9(1.010)¹¹

P = 7.698 billion people

ii) Estimate b) by what year will the population be double of that in 2019.

P = 6.9(1.010)^t

13.8 = 6.9(1.010)^t

2 = (1.010)^t

log_1.010(2) = t

t = 69.66

2019 + 69.66 = 2088.66

So, in the year 2088, the population will be double of that in 2019.

Answer 2

P = 6.9 (1.010)^t

i) You can divide both sides by 6.9 and get log from both sides: log (P/6.9) = t log(1.010) so:

t = log(1.010) / log(P/ 6.9)

ii)

a) population in year 2030 is when t is 2030 - 2019 = 11, so plug 11 to t and use calculator to get P.

P = 6.9 (1.010) ^ 11 = almost 7.698 billion in the year 2030

b) Plug t = 0 to get the population in 2019. So: P = 6.9 (1.010)^0 = 6.6 billion.

We plug 2 (6.9) into the original equation and get log to find t.

2 (6.9) = 6.9 (1.010)^t, divide both sides by 6.9, so:

2 = (1.101)^t Now get log from both sides:

log(2) = t log(1.010)

t = log(2) / log(1.010) = 69.66

Meaning after almost 69.66 years, from 2019 which is the year 2088, the population will be twice as much as it is in 2019, which are the same answers that the previous tutor got, but a little bit different approach and explanation.