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 \( \log_{175}(5x) = \log_{343}(7x) \), we will first rewrite the logarithms in terms of natural logarithms or common logarithms using the change of base formula.

### Step 1: Rewrite the logarithmic equations
Using the change of base formula, we can express the logarithms as follows:

\[ \log_{175}(5x) = \frac{\log(5x)}{\log(175)} \]
\[ \log_{343}(7x) = \frac{\log(7x)}{\log(343)} \]

Setting these two expressions equal gives us:

\[ \frac{\log(5x)}{\log(175)} = \frac{\log(7x)}{\log(343)} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:

\[ \log(5x) \cdot \log(343) = \log(7x) \cdot \log(175) \]

### Step 3: Expand the logarithms
Using the property of logarithms \( \log(ab) = \log(a) + \log(b) \), we can expand both sides:

\[ (\log(5) + \log(x)) \cdot \log(343) = (\log(7) + \log(x)) \cdot \log(175) \]

### Step 4: Distribute the logarithms
Distributing gives us:

\[ \log(5) \cdot \log(343) + \log(x) \cdot \log(343) = \log(7) \cdot \log(175) + \log(x) \cdot \log(175) \]

### Step 5: Rearranging the equation
Now, we can rearrange the equation to isolate \( \log(x) \):

\[ \log(x) \cdot \log(343) - \log(x) \cdot \log(175) = \log(7) \cdot \log(175) - \log(5) \cdot \log(343) \]

Factoring out \( \log(x) \):

\[ \log(x) \cdot (\log(343) - \log(175)) = \log(7) \cdot \log(175) - \log(5) \cdot \log(343) \]

### Step 6: Solve for \( \log(x) \)
Now, we can solve for \( \log(x) \):

\[ \log(x) = \frac{\log(7) \cdot \log(175) - \log(5) \cdot \log(343)}{\log(343) - \log(175)} \]

### Step 7: Calculate \( x \)
To find \( x \), we need to exponentiate both sides:

\[ x = 10^{\frac{\log(7) \cdot \log(175) - \log(5) \cdot \log(343)}{\log(343) - \log(175)}} \]

### Step 8: Find \( \log_{42}(x^4 - 2x^2 + 7) \)
Now we need to find \( \log_{42}(x^4 - 2x^2 + 7) \).

Notice that \( x^4 - 2x^2 + 7 \) can be rewritten as:

\[ x^4 - 2x^2 + 1 + 6 = (x^2 - 1)^2 + 6 \]

### Step 9: Substitute \( x \) into the expression
We can substitute the value of \( x \) into the expression \( (x^2 - 1)^2 + 6 \) and then calculate \( \log_{42}((x^2 - 1)^2 + 6) \).

### Final Step: Evaluate the logarithm
Finally, we can evaluate \( \log_{42}((x^2 - 1)^2 + 6) \) to get the final answer.