`ln ab-ln|b|=`

Options

A. `ln a`
B. `ln|a|`
C. `-lna`
D. none of these

Answer

To solve the equation \( \ln(ab) - \ln|b| \), we can use the properties of logarithms. Here’s a step-by-step solution:

### Step 1: Apply the logarithmic property
We know that the difference of logarithms can be expressed as the logarithm of a quotient. Therefore, we can write:

\[ \ln(ab) - \ln|b| = \ln\left(\frac{ab}{|b|}\right) \]

### Step 2: Simplify the expression
Next, we simplify the expression inside the logarithm. Since \( b \) can be either positive or negative, we can consider the absolute value:

\[ \frac{ab}{|b|} = a \cdot \frac{b}{|b|} \]

### Step 3: Analyze the fraction
The term \( \frac{b}{|b|} \) is equal to \( 1 \) if \( b > 0 \) and \( -1 \) if \( b < 0 \). Therefore, we can express this as:

\[ \frac{b}{|b|} = \text{sgn}(b) \]

Where \( \text{sgn}(b) \) is the sign function, which gives \( 1 \) for positive \( b \) and \( -1 \) for negative \( b \).

### Step 4: Final expression
Thus, we have:

\[ \ln(ab) - \ln|b| = \ln(a \cdot \text{sgn}(b)) \]

### Conclusion
So, the final result is:

\[ \ln(ab) - \ln|b| = \ln(a) + \ln(\text{sgn}(b)) \]