Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to $\ln \left(\frac{3}{4}\right)$
Write the first four nonzero terms for the series in order.
\[
\ln \left(\frac{3}{4}\right)=(\square)+(\square)+(\square)+(\square)+\cdots
\]
(Simplify your answers.)

Step 1 ：Given the Taylor series for the natural logarithm function, ln(x), centered at a=1 is: \[\ln(x) = (x-1) - \frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3 - \frac{1}{4}(x-1)^4 + \cdots\]

Step 2 ：We can use this series to approximate ln(3/4) by substituting x=3/4 into the series. The first four nonzero terms of the series will be: \[\ln \left(\frac{3}{4}\right) = \left(\frac{3}{4}-1\right) - \frac{1}{2}\left(\frac{3}{4}-1\right)^2 + \frac{1}{3}\left(\frac{3}{4}-1\right)^3 - \frac{1}{4}\left(\frac{3}{4}-1\right)^4\]

Step 3 ：Simplify these terms to get: term1 = -0.25, term2 = -0.03125, term3 = -0.005208333333333333, term4 = -0.0009765625

Step 4 ：Final Answer: The first four nonzero terms of the series that is equal to \(\ln \left(\frac{3}{4}\right)\) are \(\boxed{-0.25}\), \(\boxed{-0.03125}\), \(\boxed{-0.005208333333333333}\), and \(\boxed{-0.0009765625}\).