Exponential and Logarithmic Models

Can someone explain how to do this step by step.
The table shows the population (in millions) of a country in 1997 and the projected population (in millions) for the year 2020.

Country 1997 2020
Croatia 5.0 4.8
Mali 9.9 20.4
Singapore 3.5 4.3
Sweden 8.9 9.5

1. Find the exponential growth model y=ae^bt for the population in each country by letting t=0 correspond to 1997. Use the model to estimate the population of each country in 2030

At t=0, e^0=1, so "a" is always the number in the 1997 column.
Years are from 1997, so t=23 in 2020, and t=33 in 2030.

I'll do Croatia for you:

y = ae^bt
but we know a=5.0, and we are given y=4.8 when t=23:
=> 4.8 = 5.0 e^23b
=> ln(4.8) = ln(5.0 e^23b)
=> ln(4.8) = ln(5.0) + ln(e^23b)
=> ln(4.8) = ln(5.0) + 23b
... because ln and e^ are inverses of each other
=> 23b = ln(4.8) - ln(5.0)
=> b = (ln(4.8) - ln(5.0)) / 23 = -0.00177

For estimated population in 2030, set t=33:
pop = a e^33b
= 5.0 e^(33 x -0.00177)
= 5.0 e^(-0.0586)
= 4.716

Do others in the same way. Notice Croatia's population is decreasing so I got a negative value for b. All the others will give positive values for b.