How do you find the exponential function that contains both of these points (2,12.6) and (5, 42.525)?

Jun 18, 2015

Suppose $f \left(x\right) = k \cdot {a}^{x}$ for some $k \in \mathbb{R}$ and $a > 0$, $a \ne 1$

Then $a = \sqrt[3]{\frac{42.525}{12.6}} = \sqrt[3]{3.375} = 1.5$

and $k = \frac{12.6}{{a}^{2}} = \frac{12.6}{2.25} = 5.6$

So $f \left(x\right) = 5.6 {\left(1.5\right)}^{x}$

Explanation:

Suppose $f \left(x\right) = k \cdot {a}^{x}$ for some $k \in \mathbb{R}$ and $a > 0$, $a \ne 1$

Then $f \frac{5}{f} \left(2\right) = \frac{k \cdot {a}^{5}}{k \cdot {a}^{2}} = {a}^{3}$

So $a = \sqrt[3]{f \frac{5}{f} \left(2\right)} = \sqrt[3]{\frac{42.525}{12.6}} = \sqrt[3]{3.375} = 1.5$

$f \left(2\right) = k \cdot {a}^{2}$, so $k = f \frac{2}{{a}^{2}} = \frac{12.6}{{1.5}^{2}} = \frac{12.6}{2.25} = 5.6$

So $f \left(x\right) = 5.6 {\left(1.5\right)}^{x}$