Question

If f is an exponential function and f(x)=ab^x and f contains (2,3) and (4,5), the. B =

Answer 1

Step 1

`f(x)=ab^x`

when x = 2, f(2) = 3

`\begin{gathered} f(2)=ab^2 \\ ab^2\text{ = 3 ------------------------------ (1)} \end{gathered}`

when x = 4, f(4) = 5

`\begin{gathered} f(4)=ab^4 \\ ab^4\text{ = 5 ------------------------- (2)} \end{gathered}`

Step 2: Solve equations 1 and 2 simultaneously to find the value of a and b.

`\begin{gathered} \text{From equation 1, make b}^2\text{ subject of relation and substitute} \\ in\text{equation 2} \end{gathered}`

Therefore

`\begin{gathered} \text{From ab}^2\text{ = 3} \\ b^2\text{ = }(3)/(a) \end{gathered}`
`\begin{gathered} \text{From equation 2} \\ ab^4\text{ = 5} \\ a\text{ x (}(3)/(a))^2\text{ = 5} \\ a\text{ x }(9)/(a^2)\text{ = 5} \\ (9)/(a)\text{ = 5} \\ \text{Cross multiply} \\ 5a\text{ = 9} \\ a\text{ = }(9)/(5) \end{gathered}`

Step 3: Substitute a in equation 1 to find the value of b.

`\begin{gathered} b^2\text{ = }(3)/(a) \\ b^2\text{ = 3 }(.)/(.)\text{ a} \\ b^2\text{ = 3 }(.)/(.)\text{ }(9)/(5) \\ b^2\text{ = 3 x }(5)/(9) \\ b^2\text{ = }(5)/(3) \\ \text{b = }\sqrt[]{(5)/(3)} \end{gathered}`

Final answer

`b\text{ = }\sqrt[]{(5)/(3)}`