You're looking for the time \(t\) that it takes for the principal \(P\) to double, so \(A=2P\). Dividing both sides of the formula by \(P\) leaves you with
\(2=\left(1+\dfrac rn\right)^{nt}\)
You're given that the account earns 0.75% interest and that the interest is compounded monthly, so \(r=0.0075\) and \(n=12\).
\(2=\left(1+\dfrac{0.0075}{12}\right)^{12t}\)
\(\ln2=\ln\left(1+\dfrac{0.0075}{12}\right)^{12t}\)
\(\ln2=12t\ln\left(1+\dfrac{0.0075}{12}\right)\)
\(t=\dfrac{\ln2}{12\ln\left(1+\dfrac{0.0075}{12}\right)}\approx92.45\)
