Find the solution of the exponential equation \[ 100(1.05)^{2 t}=500,000 \] in terms of logarithms, or correct to four decimal places. \[ t= \]

Step 1 ：Divide both sides of the equation by 100 to get \(1.05^{2t} = 5000\)

Step 2 ：Take the natural logarithm (ln) of both sides of the equation to get \(2t \cdot \ln(1.05) = \ln(5000)\)

Step 3 ：Divide both sides of the equation by \(2 \cdot \ln(1.05)\) to solve for t, which gives \(t = \frac{\ln(5000)}{2 \cdot \ln(1.05)}\)

Step 4 ：Calculate the value of t to get \(t \approx 87.2839\)

Step 5 ：\(\boxed{t \approx 87.2839}\)