`I flogx=(logy)/2=(logz)/5, th ten x^4y^3z^(- 2)=`

Options

A. 2
B. 10
C. 1
D. 0

Explanation

To solve the equation \( x^4 y^3 z^{-2} \) given that \( \log x = \frac{\log y}{2} = \frac{\log z}{5} \), we can follow these steps:

### Step 1: Set a common variable for logarithms
Let \( \log x = k \). Then we can express \( \log y \) and \( \log z \) in terms of \( k \):
- From \( \log x = k \), we have \( \log y = 2k \) (since \( \log y = 2 \log x \)).
- From \( \log z = 5k \) (since \( \log z = 5 \log x \)).

### Step 2: Substitute the logarithmic expressions into the equation
We need to evaluate \( x^4 y^3 z^{-2} \). Taking the logarithm of both sides, we have:
\[ \log(x^4 y^3 z^{-2}) = \log(x^4) + \log(y^3) + \log(z^{-2}) \]

### Step 3: Apply the power rule of logarithms
Using the property \( \log(a^b) = b \log a \):
\[ \log(x^4) = 4 \log x = 4k \]
\[ \log(y^3) = 3 \log y = 3(2k) = 6k \]
\[ \log(z^{-2}) = -2 \log z = -2(5k) = -10k \]

### Step 4: Combine the logarithmic expressions
Now substituting these back into our equation:
\[ \log(x^4 y^3 z^{-2}) = 4k + 6k - 10k \]
\[ = (4k + 6k - 10k) = 0 \]

### Step 5: Exponentiate to find the value of \( x^4 y^3 z^{-2} \)
Since \( \log(x^4 y^3 z^{-2}) = 0 \), we can exponentiate both sides:
\[ x^4 y^3 z^{-2} = 10^0 = 1 \]

### Final Answer
Thus, the value of \( x^4 y^3 z^{-2} \) is \( 1 \).

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