Question

-Exponential and Logarithmic functions- Write an exponential function of the form...

Answer 1

We are asked to write an exponential function of the form

`y=ab^x`

The function passes through the following two points

(3, 2) and (6, 16)

Let us substitute these two points into the above equation.

For point (3, 2):

`\begin{gathered} 2=ab^3 \\ a=(2)/(b^3) \end{gathered}`

For point (6, 16):

`\begin{gathered} 16=ab^6 \\ a=(16)/(b^6) \end{gathered}`

Equate the two equations to solve for b

`\begin{gathered} (2)/(b^3)=(16)/(b^6) \\ 2\cdot b^6=16\cdot b^3 \\ (2b^6)/(2b^3)=(16b^3)/(2b^3) \\ b^3=8 \\ b=\sqrt[3]{8} \\ b=2 \end{gathered}`

So, the value of b is 2

Finally, the value of a is given by

`a=(2)/(b^3)=(2)/((2)^3)=(2)/(8)=(1)/(4)`

So, the value of a is 1/4

Therefore, the exponential function is

`y=(1)/(4)\cdot2^x`