Question

If the exponential model f(x)=14(5)x is written with the base e, it will take the form A0ekx. What is A0 and what is k?

Answer 1

Answer:

We have:

f(x) = 14*(5)^x

And we want to write this in the form:

g(x) = A*e^(k*x)

We want to have f(x) = g(x)

This is ratter easy.

The general case is:

A*(r)^x = B*e^(k*x)

Here we have:

A = B

Then:

(r)^x = e^(k*x)

Now we can remember the relation:

(a^x)^y = a^(x*y)

Then we can write:

e^(k*x) = (e^k)^x

Then we have:

(r)^x = (e^k)^x

Then we must have:

r = e^k

Now let's go to our case:

14*(5)^x = A*e^(k*x) = A*(e^k)^x

Then we have:

14 = A

5 = (e^k)

With this equation we can find the value of k.

ln(5) = ln(e^k) = k

ln(5) = k

Then the equation g(x) is:

g(x) = 14*e^(ln(5)*x)

Answer 2

Answer:

A0=14 k +ln9

Step-by-step explanation:

To rewrite the model with the base e, we use the fact that for all b>0,

b=eln(b)

Using this property, and then the power property of logarithms, on 9x, we find that

(9)x=eln(9x)=exln(9)=e(ln(9))x

So we find that

14(9)x=14eln(9)x

So A0=14 and k=ln(9).