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