This problem requires us to solve the equation:

5^t+2 = e^2t

We need to use log_e because of the base e on the right hand side.

ln (5^t+2) = ln (e^2t)

(t + 2) ln 5 = 2t ln e

Now, ln e = 1, and we need to collect t terms together:

t ln 5 + 2 ln 5 = 2t

t (ln 5 - 2) = - 2 ln 5

So

is the required time.

Graph

We can see on the graph that the 2 curves intersect at t = 8.2, as we found above.