All exponential functions can be written in many forms. Write the function f, of, t, equals, 67000, e, start superscript, minus, 0, point, 1, 9, t, end superscriptf(t)=67000e 
−0.19t
  in the form f, of, t, equals, a, b, to the power tf(t)=ab 
t
 . Round all coefficients to four decimal places.

We are given the function \( f(t) = 67000e^{-0.19t} \), and we want to express it in the form \( f(t) = ab^t \).

### Step-by-step process:

1. The given function is in the form \( f(t) = 67000 \cdot e^{-0.19t} \).
2. We know that the natural base \( e \) can be expressed in terms of base \( b \) using the following identity:
   \[
   e^x = b^{x \cdot \ln(b)}
   \]
   To express \( e^{-0.19t} \) in terms of another base, recall that \( e^{x} = (e^{\ln(b)})^{x/\ln(b)} \), and so we multiply out the powers accordingly.

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All exponential functions can be written in many forms. Write the function f(t) = 67000e^(-0.19t) in the form f(t) = ab^t. Round all coefficients to four decimal places.

Learn how to express the exponential function f(t) = 67000e^(-0.19t) in the form f(t) = ab^t. This solution covers exponential functions, base conversion, and key formulas with coefficients rounded to four decimal places. Suitable for students in Grades 10-12.

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