So I have found this to be my formula for Newton's Law of Cooling: $ 76e^{\left(\frac{1}{2}\ln\left(\frac{59}{77}\right)t\right)} +23$
I need to figure out if my model acurately predicts when my substance reaches room temperature which is 23 degrees Celsius. Here is my working out, but I believe it is wrong because I am getting a really weird answer. I based it off a Khan Academy video.
T(t) = $ 76e^{\left(\frac{1}{2}\ln\left(\frac{59}{77}\right)t\right)} +23$
23 = $ 76e^{\left(\frac{1}{2}\ln\left(\frac{59}{77}\right)t\right)} +23$
0 = $ 76e^{\left(\frac{1}{2}\ln\left(\frac{59}{77}\right)t\right)} +23$
$0/76$ = $ e^{\left(\frac{1}{2}\ln\left(\frac{59}{77}\right)t\right)}$
ln (0) = $ {\left(\frac{1}{2}\ln\left(\frac{59}{77}\right)t\right)}$
t= $0/{\left(\frac{1}{2}\ln\left(\frac{59}{77}\right)\right)}$
t = -infitiy/ -0.13
I am sure this is not right as it is not an actual time for when my substance reaches 23 degrees or that what I think. Can someone show me how to actually solve this