Solution
Exponential expressions with natural base e are of the form
[tex]y=a.e^{kx}[/tex]
The value of 'a' is already set to be 96 since this is the coefficient of the given equation. We just need to find k
so set 3.8^x as e^kx and solve for k
[tex]\begin{gathered} 3.8^x=e^{kx} \\ 3.8^x=(e^k)^x \\ 3.8=e^k \end{gathered}[/tex]
The two sides have the same exponent of x. So the two bases are 3.8 and
e^k must be equal to
[tex]\begin{gathered} e^k=3.8 \\ k=ln(3.8) \\ k=1.335 \end{gathered}[/tex]
This means
[tex]e^{kx}=e^{1.335x}[/tex]
and we replace the
[tex]3.8^x=e^{1.335x}[/tex]
to go from
[tex]y=96(3.8)^x[/tex]
to
[tex]y=96.e^{1.335x}[/tex]
Therefore the correct answer is
[tex]y=96.e^{1.335x}[/tex]
