Question

all exponential functions can be written in many forms. write the function $f(t) = 740e^{0.29t}$ in the form $f(t) = ab^t$. round all coefficients to four decimal places.

Explanation:

Step1: Recall the property of exponents

We know that \( e^{kt}=(e^{k})^{t} \). So for the function \( f(t) = 740e^{0.29t} \), we can rewrite \( e^{0.29t} \) as \( (e^{0.29})^{t} \).

Step2: Calculate the value of \( e^{0.29} \)

Using a calculator, we find that \( e^{0.29}\approx1.3364 \) (rounded to four decimal places).

Step3: Write the function in the form \( ab^{t} \)

Substituting the value of \( e^{0.29} \) back into the function, we get \( f(t)=740\times(1.3364)^{t} \), where \( a = 740 \) and \( b\approx1.3364 \).

Answer:

\( f(t)=\boldsymbol{740\times(1.3364)^{t}} \) (or \( f(t) = 740b^{t} \) with \( b\approx1.3364 \))