The value of `(log_(a)x.log_(b)x)/(log_(a)x+log_(b)x)` is …...

Answer

To solve the expression \(\frac{(\log_a x)(\log_b x)}{(\log_a x) + (\log_b x)}\), we can use the properties of logarithms. Let's break it down step by step.

### Step 1: Rewrite the logarithms using the change of base formula
Using the change of base formula, we can express \(\log_a x\) and \(\log_b x\) in terms of natural logarithms (or any common base):
\[ \log_a x = \frac{\log x}{\log a}, \quad \log_b x = \frac{\log x}{\log b} \]

### Step 2: Substitute the values into the expression
Substituting these values into the original expression gives:
\[ \frac{\left(\frac{\log x}{\log a}\right)\left(\frac{\log x}{\log b}\right)}{\left(\frac{\log x}{\log a}\right) + \left(\frac{\log x}{\log b}\right)} \]

### Step 3: Simplify the numerator
The numerator simplifies to:
\[ \frac{(\log x)^2}{\log a \cdot \log b} \]

### Step 4: Simplify the denominator
For the denominator, we find a common denominator:
\[ \frac{\log x}{\log a} + \frac{\log x}{\log b} = \frac{\log x \cdot \log b + \log x \cdot \log a}{\log a \cdot \log b} = \frac{\log x (\log a + \log b)}{\log a \cdot \log b} \]

### Step 5: Substitute back into the expression
Now substituting back, we have:
\[ \frac{\frac{(\log x)^2}{\log a \cdot \log b}}{\frac{\log x (\log a + \log b)}{\log a \cdot \log b}} \]

### Step 6: Simplify the entire expression
The \(\log a \cdot \log b\) cancels out from the numerator and denominator:
\[ \frac{(\log x)^2}{\log x (\log a + \log b)} = \frac{\log x}{\log a + \log b} \]

### Final Result
Thus, the value of the expression \(\frac{(\log_a x)(\log_b x)}{(\log_a x) + (\log_b x)}\) simplifies to:
\[ \frac{\log x}{\log a + \log b} \]