How do you find the exponential model y=ae^(bx) that goes through the points (0,1) and (3,10)?

Mar 1, 2017

$y = {10}^{\frac{x}{3}}$

Explanation:

We know two $\left(x , y\right)$ points, so we have enough information to write a system of equations and solve for $a$ and $b$.

Equation 1:

$1 = a {e}^{0 b}$

Equation 2:

$10 = a {e}^{3 b}$

The first equation can be simplified to $a = 1$, because $0 \left(a\right)$ will always equal $0$ and any real number $x$ has the property such that ${x}^{0} = 1$.

Solve for $b$ now.

$10 = 1 \left({e}^{3 b}\right)$

$10 = {e}^{3 b}$

$\ln 10 = \ln \left({e}^{3 b}\right)$

$\ln 10 = 3 b \ln e$

$3 b = \ln 10$

$b = \frac{1}{3} \ln 10$

The function therefore has equation $y = {e}^{\frac{1}{3} \ln 10 x}$. Now let's look at simplifying the function.

Use the logarithm property ${x}^{a} = {e}^{a \ln x}$, to arrive at the result

$y = {\left({10}^{\frac{1}{3}}\right)}^{x} = {10}^{\frac{x}{3}}$

Hopefully this helps!