If `(log_(3)x)(log_(5)3)=3` , find x.

Options

A. 5
B. 9
C. 25
D. 125

Explanation

To solve the equation \((\log_{3} x)(\log_{5} 3) = 3\), we can follow these steps:

### Step 1: Rewrite the logarithms using the change of base formula
Using the change of base formula, we can express the logarithms in terms of a common base \(c\):
\[ \log_{3} x = \frac{\log_{c} x}{\log_{c} 3} \quad \text{and} \quad \log_{5} 3 = \frac{\log_{c} 3}{\log_{c} 5} \]
Substituting these into the original equation gives:
\[ \left(\frac{\log_{c} x}{\log_{c} 3}\right) \left(\frac{\log_{c} 3}{\log_{c} 5}\right) = 3 \]

### Step 2: Simplify the equation
The \(\log_{c} 3\) terms cancel out:
\[ \frac{\log_{c} x}{\log_{c} 5} = 3 \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ \log_{c} x = 3 \log_{c} 5 \]

### Step 4: Rewrite the logarithmic equation in exponential form
Using the property of logarithms, we can rewrite this as:
\[ x = 5^3 \]

### Step 5: Calculate the value of \(x\)
Calculating \(5^3\) gives:
\[ x = 125 \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{125} \]