Convert the given exponential function to the form indicated.
f(t) = 2.7(1.004)^t --> f(t) = Q0e^kt
I'm not even sure where to start with this one - any help would be greatly appreciated.
let y = f(t)/2.7 to make it easy
y = 1.004^t
ln y = t ln 1.004 = 0.00399 t
e^ln y = y = e^.00399 t
so
f(t) = 2.7 y = 2.7 e^.00399 t
To convert the given exponential function, f(t) = 2.7(1.004)^t, to the form f(t) = Q0e^kt, we need to first express the constant multiplier 1.004 as a base for the natural exponential function.
Step 1: Rewrite the exponential function as follows:
f(t) = 2.7e^(k't), where k' = ln(1.004)
Step 2: Simplify k':
Use the natural logarithm (ln) to find the value of k':
k' = ln(1.004) ≈ 0.003997
Step 3: Substitute k' into the exponential function:
f(t) = 2.7e^(0.003997t)
Now, the exponential function has been converted to the form f(t) = Q0e^kt. In this case, Q0 = 2.7 and k ≈ 0.003997.
To convert the given exponential function to the desired form, f(t) = Q0e^kt, we need to express the given function in terms of e^kt.
Let's start by rewriting the given function:
f(t) = 2.7(1.004)^t
To express this in terms of exponential form, we need to convert 1.004 to the base e.
The general formula for converting a number to the base e is:
a = e^(ln(a))
In our case, a = 1.004. So we can rewrite 1.004 as:
1.004 = e^(ln(1.004))
Now, let's substitute this back into the given function:
f(t) = 2.7 * (e^(ln(1.004)))^t
Using the exponentiation rule, we can simplify the expression inside the parentheses:
f(t) = 2.7 * e^(ln(1.004) * t)
Now, we have expressed the function in the form f(t) = Q0e^kt, with:
Q0 = 2.7
k = ln(1.004)
Therefore, the final expression is:
f(t) = 2.7e^(ln(1.004)t)
Note: In practice, we usually use the calculator to compute ln(1.004) to get an approximate value for k.