#### Answer

$s\approx140.011$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
100(1.02)^{s/4}=200
,$ divide both sides by $100.$ Then take the logarithm of both sides. Use the properties of logarithms and of equality to isolate the variable. Finally, express the answer with $3$ decimal places.
$\bf{\text{Solution Details:}}$
Dividing both sides by $100,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
(1.02)^{s/4}=\dfrac{200}{100}
\\\\
(1.02)^{s/4}=2
.\end{array}
Taking the logarithm of both sides, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\log(1.02)^{s/4}=\log2
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the equation above is equivalent
\begin{array}{l}\require{cancel}
\dfrac{s}{4}\log(1.02)=\log2
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
4\left( \dfrac{s}{4}\log(1.02) \right)=4(\log2)
\\\\
s\log(1.02)=4\log2
\\\\
s=\dfrac{4\log2}{\log(1.02)}
\\\\
s\approx140.011
.\end{array}