# How do you find the exponential function f(x)= a^x whose graph goes through the point (3, 1/125)?

Dec 6, 2016

#### Explanation:

Use the logarithm of your favorite base on the function (My favorite is the natural logarithm but base 10 works just as well):

$\ln \left(f \left(x\right)\right) = \ln \left({a}^{x}\right)$

Use the identity ln(a^x) = (x)ln(a):

$\ln \left(f \left(x\right)\right) = \left(x\right) \ln \left(a\right)$

Divide both sides by x and flip the equation:

$\ln \left(a\right) = \ln \frac{f \left(x\right)}{x}$

Make both sides exponents of the base e:

${e}^{\ln} \left(a\right) = {e}^{\ln \frac{f \left(x\right)}{x}}$

Use the identity ${e}^{\ln} \left(a\right) = a$

$a = {e}^{\ln \frac{f \left(x\right)}{x}}$

Substitute 3 for x and 1/125 for $f \left(x\right)$:

$a = {e}^{\ln \frac{\frac{1}{125}}{3}}$

$a = 0.2$

If you prefer a fraction:

$a = \frac{1}{5}$