Convert the equation f(t)=130(0.93)^t to the form f(t)=ae^kt

Question

Convert the equation ^t to the form ^kt

Answer 1

What you want is for 0.93 = e^k. Convert that form to logarithmic form, i.e. k is the exponent e is the base and 0.93 is the result of raising e^k. In other words k = ln(0.93).

Another way to look at it: if 0.93 = e^k, take ln of both sides: ln (0.93) = ln e^k = k ln(e) = k(1) = k.

So, final answer: f(t) = 130 e^ln(0.93)t . If you want you can replace ln(0.93) with a decimal approximation.

Answer 2

Hello,

There are two identities that will help you with this problem:

1. x = e^ln(x) for any real number x (except zero)
2. ln(a^b) = b ln(a)

These two identities will enable you to re-express any exponent A^b in terms of e as:

1. A^b = e^{b ln(A)}

So, (0.93)^t = e^{t ln(0.93)}

Therefore,

f(t) = 130(0.93)^t = 130e^ln(0.93)t

Where: a = 130 and k = ln(0.93)

Good luck, hope this was helpful.