If `2^(log5).5^(log2)=2^(logx) "then" log_(5) root3(x^(2))`= _____

Options

A. `4/3`
B. `3/4`
C. `1/3`
D. 3

Explanation

To solve the equation \( 2^{\log 5} \cdot 5^{\log 2} = 2^{\log x} \) and find the value of \( \log_5 \sqrt[3]{x^2} \), we can follow these steps:

### Step 1: Simplify the Left Side
We start with the expression \( 2^{\log 5} \cdot 5^{\log 2} \).

Using the property of logarithms that states \( a^{\log_b c} = c^{\log_b a} \), we can rewrite \( 5^{\log 2} \) as \( 2^{\log 5} \).

Thus, we have:
\[ 2^{\log 5} \cdot 5^{\log 2} = 2^{\log 5} \cdot 2^{\log 5} = 2^{\log 5 + \log 5} = 2^{2 \log 5} = 2^{\log 25} \]

### Step 2: Set the Exponents Equal
Now we can equate the simplified left side to the right side:
\[ 2^{\log 25} = 2^{\log x} \]

Since the bases are the same, we can set the exponents equal to each other:
\[ \log 25 = \log x \]

### Step 3: Solve for \( x \)
From the equation \( \log 25 = \log x \), we conclude:
\[ x = 25 \]

### Step 4: Find \( \log_5 \sqrt[3]{x^2} \)
Now we need to find \( \log_5 \sqrt[3]{x^2} \):
\[ \sqrt[3]{x^2} = \sqrt[3]{25^2} = \sqrt[3]{625} \]

### Step 5: Rewrite \( 625 \) in terms of \( 5 \)
Since \( 625 = 5^4 \), we can write:
\[ \sqrt[3]{625} = \sqrt[3]{5^4} = 5^{\frac{4}{3}} \]

### Step 6: Apply the Logarithm
Now we apply the logarithm:
\[ \log_5 \sqrt[3]{x^2} = \log_5 (5^{\frac{4}{3}}) = \frac{4}{3} \]

### Final Answer
Thus, the value of \( \log_5 \sqrt[3]{x^2} \) is:
\[ \frac{4}{3} \]

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