Given log x = m + n and log y = m - n, express the value of `"log" (10 x)/(y^(2))` in terms of m and n.

Answer

To solve the problem of expressing the value of \(\log \left( \frac{10x}{y^2} \right)\) in terms of \(m\) and \(n\), we will follow these steps:

### Step 1: Write the expression using logarithmic properties
We start with the expression:
\[ \log \left( \frac{10x}{y^2} \right) \]
Using the property of logarithms that states \(\log \left( \frac{a}{b} \right) = \log a - \log b\), we can rewrite the expression as:
\[ \log(10x) - \log(y^2) \]

### Step 2: Break down \(\log(10x)\) and \(\log(y^2)\)
Next, we can further break down \(\log(10x)\) using the property \(\log(ab) = \log a + \log b\):
\[ \log(10x) = \log(10) + \log(x) \]
Since \(\log(10) = 1\), we have:
\[ \log(10x) = 1 + \log(x) \]

For \(\log(y^2)\), we use the property \(\log(a^b) = b \cdot \log(a)\):
\[ \log(y^2) = 2 \log(y) \]

### Step 3: Substitute the values of \(\log(x)\) and \(\log(y)\)
From the problem, we know:
\[ \log(x) = m + n \quad \text{and} \quad \log(y) = m - n \]
Substituting these values into our expression:
\[ \log(10x) = 1 + (m + n) = 1 + m + n \]
And for \(\log(y^2)\):
\[ \log(y^2) = 2(m - n) \]

### Step 4: Combine the results
Now we can substitute these results back into our expression:
\[ \log \left( \frac{10x}{y^2} \right) = (1 + m + n) - 2(m - n) \]

### Step 5: Simplify the expression
Now we simplify:
\[ \log \left( \frac{10x}{y^2} \right) = 1 + m + n - 2m + 2n \]
Combining like terms:
\[ = 1 + (m - 2m) + (n + 2n) = 1 - m + 3n \]

### Final Answer
Thus, the value of \(\log \left( \frac{10x}{y^2} \right)\) in terms of \(m\) and \(n\) is:
\[ \log \left( \frac{10x}{y^2} \right) = 1 - m + 3n \]

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