if `(logx)/(logy)= (log 49)/(log 7)` , then the relation between x and y.

Options

A. `x = sqrty`
B. `x= y^(3)`
C. `y= x^(2)`
D. `x=y^(2)`

Explanation

To solve the problem, we start with the equation given:

\[ \frac{\log x}{\log y} = \frac{\log 49}{\log 7} \]

### Step 1: Simplify the right side
We know that \(49\) can be expressed as \(7^2\). Therefore, we can rewrite \(\log 49\) using the power rule of logarithms:

\[ \log 49 = \log(7^2) = 2 \log 7 \]

Now substituting this back into the equation gives us:

\[ \frac{\log x}{\log y} = \frac{2 \log 7}{\log 7} \]

### Step 2: Simplify the fraction
The \(\log 7\) in the numerator and denominator cancels out (assuming \(\log 7 \neq 0\)):

\[ \frac{\log x}{\log y} = 2 \]

### Step 3: Rewrite the equation in logarithmic form
Using the property of logarithms that states \(\frac{\log a}{\log b} = \log_b a\), we can rewrite the left side:

\[ \log_y x = 2 \]

### Step 4: Convert to exponential form
From the logarithmic equation \(\log_y x = 2\), we can convert it to its exponential form:

\[ x = y^2 \]

### Conclusion
Thus, the relation between \(x\) and \(y\) is:

\[ x = y^2 \]

---