If `log_(a)(ab)=x`, then `log_(b)` (ab) is equal to

Options

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

Answer

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

\[ \log_a(ab) = x \]

We need to find the value of \(\log_b(ab)\).

### Step 1: Rewrite \(\log_a(ab)\) using logarithmic properties

Using the property of logarithms that states \(\log_a(mn) = \log_a(m) + \log_a(n)\), we can rewrite \(\log_a(ab)\):

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

### Step 2: Simplify \(\log_a(a)\)

Since \(\log_a(a) = 1\), we can substitute this into our equation:

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

### Step 3: Set the equation equal to \(x\)

Now we can set this equal to \(x\):

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

### Step 4: Solve for \(\log_a(b)\)

Rearranging the equation gives us:

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

### Step 5: Change of base formula

Now, we want to find \(\log_b(ab)\). We can use the change of base formula for logarithms:

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

### Step 6: Substitute known values

We already know \(\log_a(ab) = x\) and \(\log_a(b) = x - 1\). Substituting these values gives us:

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

### Conclusion

Thus, the final result is:

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

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