The `4^(th)` term of `"log"_(e )(3)/(2)` is

Options

A. `(1)/(16)`
B. `-(1)/(16)`
C. `-(1)/(64)`
D. `(1)/(64)`

Answer

To find the 4th term of the logarithmic series for \(\log_{e}\left(\frac{3}{2}\right)\), we can follow these steps:

### Step 1: Understand the logarithmic series
The logarithmic series expansion is given by:
\[ \log(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots \]
This series converges for \(|x| < 1\).

### Step 2: Identify the value of \(x\)
In our case, we need to express \(\log_{e}\left(\frac{3}{2}\right)\) in the form of \(\log(1 + x)\). We can rewrite:
\[ \log_{e}\left(\frac{3}{2}\right) = \log_{e}(1 + \frac{1}{2}) \quad \text{(since } \frac{3}{2} = 1 + \frac{1}{2}\text{)} \]
Thus, we have:
\[ x = \frac{1}{2} \]

### Step 3: Write the terms of the series
We want to find the 4th term of the series, which corresponds to \(T_4\). The general term \(T_n\) in the series can be expressed as:
\[ T_n = (-1)^{n-1} \frac{x^n}{n} \]
For \(n = 4\):
\[ T_4 = (-1)^{4-1} \frac{x^4}{4} = -\frac{x^4}{4} \]

### Step 4: Substitute the value of \(x\)
Now substituting \(x = \frac{1}{2}\):
\[ T_4 = -\frac{\left(\frac{1}{2}\right)^4}{4} \]

### Step 5: Calculate \(\left(\frac{1}{2}\right)^4\)
Calculating \(\left(\frac{1}{2}\right)^4\):
\[ \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

### Step 6: Substitute back into the expression for \(T_4\)
Now substituting this value back:
\[ T_4 = -\frac{\frac{1}{16}}{4} = -\frac{1}{64} \]

### Conclusion
Thus, the 4th term of \(\log_{e}\left(\frac{3}{2}\right)\) is:
\[ \boxed{-\frac{1}{64}} \]