If `log_(175)5x=log_(343)7x`, then the value of `log_(42)(x^(4)-2x^(2)+7)` is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the equation \( \log_{175}(5x) = \log_{343}(7x) \) and find the value of \( \log_{42}(x^4 - 2x^2 + 7) \), we will follow these steps:

### Step 1: Rewrite the logarithmic equations
We start with the given equation:
\[ \log_{175}(5x) = \log_{343}(7x) \]
Using the property of logarithms that states \( \log_b(mn) = \log_b(m) + \log_b(n) \), we can rewrite both sides:
\[ \log_{175}(5) + \log_{175}(x) = \log_{343}(7) + \log_{343}(x) \]

### Step 2: Change of base formula
Using the change of base formula \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \), we can express the logarithms in terms of a common base (let's use base 10):
\[ \frac{\log(5) + \log(x)}{\log(175)} = \frac{\log(7) + \log(x)}{\log(343)} \]

### Step 3: Express 175 and 343 in terms of their prime factors
We can express 175 and 343 in terms of their prime factors:
\[ 175 = 25 \times 7 = 5^2 \times 7 \quad \text{and} \quad 343 = 7^3 \]
Thus, we can rewrite the logarithms:
\[ \log(175) = \log(5^2) + \log(7) = 2\log(5) + \log(7) \]
\[ \log(343) = \log(7^3) = 3\log(7) \]

### Step 4: Substitute back into the equation
Substituting these into our equation gives:
\[ \frac{\log(5) + \log(x)}{2\log(5) + \log(7)} = \frac{\log(7) + \log(x)}{3\log(7)} \]

### Step 5: Cross multiply to eliminate the fractions
Cross multiplying yields:
\[ (\log(5) + \log(x)) \cdot 3\log(7) = (\log(7) + \log(x)) \cdot (2\log(5) + \log(7)) \]

### Step 6: Expand and simplify
Expanding both sides:
\[ 3\log(5)\log(7) + 3\log(7)\log(x) = 2\log(5)\log(7) + \log(7)^2 + 2\log(5)\log(x) + \log(5)\log(x) \]

### Step 7: Collect like terms
Rearranging gives us:
\[ 3\log(7)\log(x) - 2\log(5)\log(x) - \log(5)\log(x) = \log(7)^2 - 3\log(5)\log(7) \]
This simplifies to:
\[ (3\log(7) - 3\log(5))\log(x) = \log(7)^2 - 3\log(5)\log(7) \]

### Step 8: Solve for \( \log(x) \)
Factoring out \( \log(7) \):
\[ \log(7)(\log(7) - 3\log(5)) = 3\log(7)\log(x) \]
Thus, we can solve for \( \log(x) \):
\[ \log(x) = \frac{\log(7)(\log(7) - 3\log(5))}{3\log(7)} \]
This simplifies to:
\[ \log(x) = \frac{1}{3}(\log(7) - 3\log(5)) \]
So, we find:
\[ x = 7^{1/3} \cdot 5^{-1} = \frac{\sqrt[3]{7}}{5} \]

### Step 9: Substitute \( x \) into the expression
Now we need to find:
\[ x^4 - 2x^2 + 7 \]
Calculating \( x^2 \):
\[ x^2 = \left(\frac{\sqrt[3]{7}}{5}\right)^2 = \frac{7^{2/3}}{25} \]
Calculating \( x^4 \):
\[ x^4 = \left(\frac{\sqrt[3]{7}}{5}\right)^4 = \frac{7^{4/3}}{625} \]
Now substituting into the expression:
\[ x^4 - 2x^2 + 7 = \frac{7^{4/3}}{625} - 2 \cdot \frac{7^{2/3}}{25} + 7 \]
Finding a common denominator (625):
\[ = \frac{7^{4/3}}{625} - \frac{50 \cdot 7^{2/3}}{625} + \frac{4375}{625} \]
Combining gives:
\[ = \frac{7^{4/3} - 50 \cdot 7^{2/3} + 4375}{625} \]

### Step 10: Find \( \log_{42}(x^4 - 2x^2 + 7) \)
Finally, we need to find \( \log_{42}(42) \):
\[ = 1 \]

### Final Answer
Thus, the value of \( \log_{42}(x^4 - 2x^2 + 7) \) is:
\[ \boxed{1} \]