Pa^t and Pe^rt are two forms of the same exponential function.
One would expect to obtain the same answer for f(18).
Explain why you may not get the same answer for f(18) in practice.
Pa^t=Pe^rt
take the ln of each side.
tlna=rt
lna=r
If you don't have ln (a)=r, you cannot get the same answer.
In theory, both Pa^t and Pe^rt represent the same exponential function, where 'P' is the initial value, 'a' or 'e' is the base of the exponential, 'r' is the growth or decay rate, and 't' is the time. Both forms should give the same answer for any given value of 't'. However, in practice, you may not get the same answer for f(18) due to a few reasons:
1. Precision: When performing calculations with real numbers, especially with irrational numbers like 'e', there can be rounding errors or limited precision. Computers and calculators have a finite number of digits they can handle, so there might be slight differences in the values calculated using 'a' and 'e', resulting in slightly different outputs for f(18).
2. Approximations: In some cases, particularly when working with 'e', calculations might involve approximations or rounding. Different methods of approximation or rounding can lead to slight variations in the results obtained using 'a' and 'e', causing the outputs for f(18) to differ.
3. Implementation differences: The algorithms or programming used to compute the values of f(18) using 'a' and 'e' can vary. Different algorithms may introduce slight variations or differences in calculation methods, which can affect the final results obtained.
While these differences may be negligible or minor, they can still result in different answers for f(18) when computing with 'Pa^t' and 'Pe^rt' in practice.