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 = 2tlne

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

tln 5 + 2 ln 5 = 2t

t(ln 5 − 2) = −2 ln 5

So

`t=(-2\ ln\ 5)/(ln\ 5-2)=8.241649476`

is the required time.

**Graph**

Graphs of `y=e^(2t)` (magenta) and `y=5^(t+2)` (green) showing the intersection point.

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

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