If `log(x+4)=log(4)+log(x)` and `log(y+6)=logy+log6` , then which of the following is correct?

Options

A. `x=y`
B. `xlty`
C. `xgty`
D. cant say

Answer

To solve the equations given in the question, we will follow these steps:

### Step 1: Solve the first equation
The first equation is:
\[ \log(x + 4) = \log(4) + \log(x) \]

Using the property of logarithms that states \( \log(a) + \log(b) = \log(ab) \), we can rewrite the right side:
\[ \log(x + 4) = \log(4x) \]

Since the logarithm function is one-to-one, we can equate the arguments:
\[ x + 4 = 4x \]

### Step 2: Rearrange the equation
Now, we will rearrange the equation to isolate \( x \):
\[ x + 4 = 4x \]
Subtract \( x \) from both sides:
\[ 4 = 4x - x \]
This simplifies to:
\[ 4 = 3x \]

### Step 3: Solve for \( x \)
Now, divide both sides by 3 to find \( x \):
\[ x = \frac{4}{3} \]

### Step 4: Solve the second equation
The second equation is:
\[ \log(y + 6) = \log(y) + \log(6) \]

Again, using the property of logarithms:
\[ \log(y + 6) = \log(6y) \]

Equating the arguments gives us:
\[ y + 6 = 6y \]

### Step 5: Rearrange the second equation
Rearranging this equation:
\[ y + 6 = 6y \]
Subtract \( y \) from both sides:
\[ 6 = 6y - y \]
This simplifies to:
\[ 6 = 5y \]

### Step 6: Solve for \( y \)
Now, divide both sides by 5 to find \( y \):
\[ y = \frac{6}{5} \]

### Step 7: Compare \( x \) and \( y \)
Now we have:
\[ x = \frac{4}{3} \approx 1.333 \]
\[ y = \frac{6}{5} = 1.2 \]

Since \( x \) is greater than \( y \):
\[ x > y \]

### Conclusion
The correct option based on the comparison of \( x \) and \( y \) is that \( x \) is greater than \( y \).

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