If `"log"_(a) ab = x,` then the value of `"log"_(b)ab,` is

Options

A. `(x-1)/(x)`
B. `(x)/(x-1)`
C. `(x)/(x+1)`
D. `(x+1)/(x)`

Answer

To solve the problem, we start with the given equation:

\[ \log_a(ab) = x \]

### Step 1: Rewrite the logarithm using properties
Using the property of logarithms, we can expand \(\log_a(ab)\):

\[ \log_a(ab) = \log_a(a) + \log_a(b) \]

### Step 2: Substitute known values
We know that \(\log_a(a) = 1\), so we can substitute this value into our equation:

\[ 1 + \log_a(b) = x \]

### Step 3: Isolate \(\log_a(b)\)
Now, we isolate \(\log_a(b)\):

\[ \log_a(b) = x - 1 \]

### Step 4: Change of base formula
Next, we want to find \(\log_b(ab)\). We can use the change of base formula:

\[ \log_b(ab) = \frac{\log_a(ab)}{\log_a(b)} \]

### Step 5: Substitute values into the change of base formula
We already know \(\log_a(ab) = x\) and \(\log_a(b) = x - 1\):

\[ \log_b(ab) = \frac{x}{x - 1} \]

### Final Result
Thus, the value of \(\log_b(ab)\) is:

\[ \log_b(ab) = \frac{x}{x - 1} \]

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