Solve : `log_(3)x . log_(4)x.log_(5)x=log_(3)x.log_(4)x+log_(4)x.log_(5)+log_(5)x.log_(3)x`.

Explanation

To solve the equation

\[ \log_{3} x \cdot \log_{4} x \cdot \log_{5} x = \log_{3} x \cdot \log_{4} x + \log_{4} x \cdot \log_{5} x + \log_{5} x \cdot \log_{3} x, \]

we can follow these steps:

### Step 1: Let \( \log_{3} x = p \), \( \log_{4} x = q \), and \( \log_{5} x = r \).

This substitution simplifies our equation.

### Step 2: Rewrite the equation using \( p, q, r \).

The equation now becomes:

\[ pqr = pq + qr + rp. \]

### Step 3: Rearrange the equation.

We can rearrange this to:

\[ pqr - (pq + qr + rp) = 0. \]

### Step 4: Factor the equation.

This can be factored as:

\[ pqr - pq - qr - rp = 0 \implies \frac{1}{pqr} = \frac{1}{pq} + \frac{1}{qr} + \frac{1}{rp}. \]

### Step 5: Take the common denominator.

Taking the common denominator on the right-hand side gives us:

\[ \frac{1}{pqr} = \frac{r}{pqr} + \frac{p}{pqr} + \frac{q}{pqr}. \]

### Step 6: Cancel \( \frac{1}{pqr} \) from both sides.

This simplifies to:

\[ 1 = p + q + r. \]

### Step 7: Substitute back the logarithmic expressions.

Now substituting back the values of \( p, q, r \):

\[ \log_{3} x + \log_{4} x + \log_{5} x = 1. \]

### Step 8: Combine the logarithms.

Using the property of logarithms, we can combine these:

\[ \log_{3}(x) + \log_{4}(x) + \log_{5}(x) = \log_{3}(x) + \log_{3}(x) + \log_{3}(x) \cdot \frac{\log_{4}(x)}{\log_{3}(4)} + \log_{3}(x) \cdot \frac{\log_{5}(x)}{\log_{3}(5)} = 1. \]

### Step 9: Convert to a single logarithm.

This implies:

\[ \log_{3}(x) + \log_{4}(x) + \log_{5}(x) = \log_{3}(x) + \log_{3}(x) + \log_{3}(x) \cdot \frac{\log_{4}(x)}{\log_{3}(4)} + \log_{3}(x) \cdot \frac{\log_{5}(x)}{\log_{3}(5)} = 1. \]

### Step 10: Solve for x.

This can be rewritten as:

\[ \log_{3}(x) + \log_{4}(x) + \log_{5}(x) = \log_{3}(60). \]

Thus, we have:

\[ \log_{3}(60) = 1 \implies x = 60. \]

### Final Answer:

Therefore, the solution is

\[ \boxed{60}. \]

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