1960, the u. s. per capita personal income was $2283. in 2000, it was $30,069. find an exponential function to model this data

f(t) = 2283·(30069/2283)^(t/40) . . . . . t = years after 1960

In simplest terms, the exponential function can be written from the initial value, the ratio of given values, and the time period over which that ratio was effective. The form is ...

... f(t) = (initial value) · (ratio of values)^(t/(time period))

This works for both increasing and decreasing exponentials.

Using e as a base

It can be converted to an exponential with "e" as the base by taking logarithms.

ln(f(t)) = ln(2283) + (t/40)·ln(30069/2283) = ln(2283) + 0.06445011·t

Taking antilogs, this is ...

... f(t) = 2283·e^(0.06445011·t)

Comment on accuracy

The final number (30,069) when including cents (30,069.00) has 7 significant digits. In order to get the function f(t) to reproduce that number to 7 significant digits, the multiplier of t in the exponential function must be accurate to 7 significant digits. (Fairly commonly, you will see it rounded to 2 or 3 significant digits. It cannot give 30069 even to 5 digits in that case.)

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