Put the function P=15(1.1)^t in the form Pe^kt
To rewrite the exponential function P=15(1.1)^t in the form Pe^kt, we need to find the values of P, k, and e.
First, let's rewrite the given function as P = 15 * (1.1)^t.
Comparing this with the form Pe^kt, we can see that P is equivalent to 15, e is equivalent to 1.1, and k is unknown.
To find k, we can take the natural logarithm (ln) of both sides of the equation:
ln(P) = ln(15 * (1.1)^t)
Applying the logarithmic property, we get:
ln(P) = ln(15) + ln(1.1^t)
Using the power rule of logarithms, we have:
ln(P) = ln(15) + t * ln(1.1)
Now, we have our equation in the form:
ln(P) = k * t + ln(P0), where k is equivalent to ln(1.1) and ln(P0) is equivalent to ln(15).
To summarize, the function P = 15(1.1)^t can be expressed in the form Pe^kt as:
P = e^(ln(P)) = e^(k * t + ln(P0)),
where k ≈ ln(1.1) and P0 ≈ 15.
To put the function P = 15(1.1)^t in the form Pe^kt, we need to express the exponential term (1.1)^t in terms of the base of the natural logarithm, which is e.
To do this, we can rewrite the exponential term as e raised to the power of the natural logarithm of 1.1.
We know that ln(a^b) = b * ln(a), so in our case, ln(1.1^t) = t * ln(1.1).
Now, let's rewrite the function using this information:
P = 15(1.1)^t
P = 15(e^(ln(1.1)^t))
P = 15e^(t * ln(1.1))
Thus, the function P = 15(1.1)^t is in the form Pe^kt, where k = ln(1.1).