The amount of outstanding consumer debt in trillions of doll

The amount of outstanding consumer debt (in trillions of dollars) is approximated by g(t) = 0.365132 1.076502^t, where t = 0 corresponds to 1980. Find the year in which consumer debt is $2 trillion $9 trillion

Solution

g(t) = 0.365132*( 1.076502)^t

t = 0 corresponds to 1980

Find t when g(t) = $2 trillion

2 = 0.365132*( 1.076502)^t

5.47747 = 1.076502^t

Taking log of both sides:

log5.47747 = t*log( 1.076502)

t = 23.06 corresponding to year 2004 as 23 corresponds to year 2003

b)   Find t when g(t) = $9 trillion

9 =  0.365132*( 1.076502)^t

24.6486 =( 1.076502)^t

Taking log on both sides:

log24.6486 = t*log1.076502

t = 43.47 which corresponds to year 2024 as 43 corresponds to year 2023