Question

Express the answer in terms of a natural logarithm. Y= (do not simplify)

Answer 1

Exponential expressions with natural base e are of the form

`y=a.e^(kx)`

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

`\begin{gathered} 3.8^x=e^(kx) \\ 3.8^x=(e^k)^x \\ 3.8=e^k \end{gathered}`

The two sides have the same exponent of x. So the two bases are 3.8 and

e^k must be equal to

`\begin{gathered} e^k=3.8 \\ k=ln(3.8) \\ k=1.335 \end{gathered}`

This means

`e^(kx)=e^(1.335x)`

and we replace the

`3.8^x=e^(1.335x)`

to go from

`y=96(3.8)^x`

to

`y=96.e^(1.335x)`

Therefore the correct answer is

`y=96.e^(1.335x)`