2log x-log(x+1)-log(x-1) is equals to

Options

A. `x^(2)+1/2x^(4)+1/3x^(6)`+..
B. `(1)/(x^(2))+(1)/(2x^(4))+(1)/(3x^(6))`+..
C. `-{(1)/(x^(2))+(1)/(2x^(4))+(1)/(3x^(6))`+..
D. `-(1)/(n)(w)^(n)+w^(2n)`

Answer

To solve the equation \( 2\log x - \log(x+1) - \log(x-1) \), we will use properties of logarithms step by step.

### Step 1: Apply the properties of logarithms
We know that \( a \log b = \log(b^a) \) and \( \log a - \log b = \log\left(\frac{a}{b}\right) \).

Using these properties, we can rewrite the expression:
\[ 2\log x = \log(x^2) \]
Thus, we can rewrite the entire expression as:
\[ \log(x^2) - \log(x+1) - \log(x-1) \]

### Step 2: Combine the logarithms
Now, we can combine the logarithms using the property \( \log a - \log b = \log\left(\frac{a}{b}\right) \):
\[ \log\left(\frac{x^2}{(x+1)(x-1)}\right) \]

### Step 3: Simplify the fraction
Next, we simplify the fraction inside the logarithm:
\[ (x+1)(x-1) = x^2 - 1 \]
So we have:
\[ \log\left(\frac{x^2}{x^2 - 1}\right) \]

### Step 4: Rewrite the logarithm
Using the property of logarithms, we can express this as:
\[ \log\left(x^2\right) - \log\left(x^2 - 1\right) \]

### Step 5: Further simplification
We can also express this in terms of a single logarithm:
\[ \log\left(\frac{x^2}{x^2 - 1}\right) \]

### Final Result
Thus, the final expression is:
\[ \log\left(\frac{x^2}{x^2 - 1}\right) \]