Find the value of x and y respectively for`log_10(x^2y^3)=7` and `log_10(x//y)=1`:

Options

A. x=10,y=100
B. x=100,y=10
C. x=10,y=20
D. none of these

Answer

To solve the equations \( \log_{10}(x^2y^3) = 7 \) and \( \log_{10}\left(\frac{x}{y}\right) = 1 \), we will follow these steps:

### Step 1: Convert the logarithmic equations to exponential form

From the second equation:
\[ \log_{10}\left(\frac{x}{y}\right) = 1 \]
Using the property of logarithms, we can convert this to:
\[ \frac{x}{y} = 10^1 \]
Thus, we have:
\[ \frac{x}{y} = 10 \quad \text{(1)} \]

### Step 2: Express \( x \) in terms of \( y \)

From equation (1), we can express \( x \) in terms of \( y \):
\[ x = 10y \quad \text{(2)} \]

### Step 3: Substitute \( x \) into the first equation

Now, we substitute equation (2) into the first equation:
\[ \log_{10}(x^2y^3) = 7 \]
Substituting \( x = 10y \):
\[ \log_{10}((10y)^2y^3) = 7 \]
This simplifies to:
\[ \log_{10}(100y^2y^3) = 7 \]
\[ \log_{10}(100y^5) = 7 \]

### Step 4: Use the property of logarithms again

Using the property of logarithms:
\[ \log_{10}(100) + \log_{10}(y^5) = 7 \]
Since \( \log_{10}(100) = 2 \):
\[ 2 + \log_{10}(y^5) = 7 \]
Subtracting 2 from both sides:
\[ \log_{10}(y^5) = 5 \]

### Step 5: Convert back to exponential form

Now, converting back to exponential form:
\[ y^5 = 10^5 \]
Taking the fifth root of both sides:
\[ y = 10 \]

### Step 6: Find \( x \) using \( y \)

Now that we have \( y \), we can find \( x \) using equation (2):
\[ x = 10y = 10 \times 10 = 100 \]

### Final Answer

Thus, the values of \( x \) and \( y \) are:
\[ x = 100, \quad y = 10 \]