Question

All exponential functions can be written in many forms. Write the function

�
(
�
)
=
10000
(
1.25
)
�
2
f(t)=10000(1.25)
2
t
​

in the form
�
(
�
)
=
�
�
�
�
f(t)=ae
kt
. Round all coefficients to four decimal places.

Answer 1

To write the function f(t)=10000(1.25)^(2t) in the form f(t)=ae^(kt), we need to take the natural logarithm of both sides and use the properties of logarithms to simplify the expression.

f(t) = 10000(1.25)^(2t)

ln(f(t)) = ln(10000(1.25)^(2t))

ln(f(t)) = ln(10000) + ln(1.25^(2t))

ln(f(t)) = ln(10000) + 2t ln(1.25)

Now, we can see that this is in the form of f(t) = ae^(kt), where a = ln(10000), k = ln(1.25), and e is the natural base. Rounding to four decimal places, we get:

f(t) = 9.2103e^(0.2231t)