Determining the parts of an exponential equation based on the graph.

Question

I am given the following equation and asked to find $a$

$$s(t) = Ae^{-at}.$$

I know $A = 9$ and I know at $t = 1, s(1) = 3$ so I create the following equation

$$3 = 9e^{-a.1}$$

and reduce it down to

$$\frac{1}{3} = e^{-a}.$$

I then get rid of the $e$ by using $\ln$ and get

$$\ln\left(\frac{1}{3}\right) = -a$$

so I get the following answer

$$-\ln\left(\frac{1}{3}\right) = a.$$

However WolframAlpha and my teacher list it as $a = \ln(3)$ what did I miss?

Answer 1

One of the 'log laws' is $\ln(a^b) = b\ln a$ for $a > 0$. Note that $$\ln\left(\frac{1}{3}\right) = \ln(3^{-1}) = -\ln(3)$$ so $-\ln\left(\frac{1}{3}\right) = \ln(3)$.