`(1 + log_(n)m) * log_(mn) x` is equal to

Options

A. `log_(n)x`
B. `log_(m)x`
C. `log_(n)m`
D. `log_(x)n`

Answer

To solve the expression \((1 + \log_{n} m) \cdot \log_{mn} x\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (1 + \log_{n} m) \cdot \log_{mn} x \]
We can rewrite \(1\) as \(\log_{n} n\) because \(\log_{n} n = 1\):
\[ (\log_{n} n + \log_{n} m) \cdot \log_{mn} x \]

### Step 2: Use the property of logarithms
Using the property of logarithms that states \(\log_{a} b + \log_{a} c = \log_{a} (bc)\), we can combine the logarithms:
\[ \log_{n} (n \cdot m) \cdot \log_{mn} x \]

### Step 3: Change the base of the logarithm
Next, we need to express \(\log_{mn} x\) in terms of base \(n\). We can use the change of base formula:
\[ \log_{mn} x = \frac{\log_{n} x}{\log_{n} (mn)} \]
We know that \(\log_{n} (mn) = \log_{n} m + \log_{n} n\):
\[ \log_{mn} x = \frac{\log_{n} x}{\log_{n} m + 1} \]

### Step 4: Substitute back into the expression
Now we substitute \(\log_{mn} x\) back into our expression:
\[ \log_{n} (n \cdot m) \cdot \frac{\log_{n} x}{\log_{n} m + 1} \]

### Step 5: Simplify the expression
Now, we can simplify:
\[ \log_{n} (n \cdot m) = \log_{n} n + \log_{n} m = 1 + \log_{n} m \]
Thus, our expression becomes:
\[ (1 + \log_{n} m) \cdot \frac{\log_{n} x}{\log_{n} m + 1} \]
The \((1 + \log_{n} m)\) terms cancel out:
\[ \log_{n} x \]

### Final Result
Thus, the final result is:
\[ \log_{n} x \]