If `log(x-5)=log(x)-log(5)` and `log(y-6)=logy-log6` , then which of the following is correct ?

Options

A. `xgty`
B. `xlty`
C. `x=y`
D. can't say

Answer

To solve the equations given in the question, we will use the properties of logarithms. Let's break down each equation step by step.

### Step 1: Solve the first equation `log(x - 5) = log(x) - log(5)`

Using the property of logarithms that states `log(a) - log(b) = log(a/b)`, we can rewrite the right side of the equation:

\[ log(x - 5) = log\left(\frac{x}{5}\right) \]

### Step 2: Set the arguments of the logarithms equal to each other

Since the logarithm function is one-to-one, we can set the arguments equal to each other:

\[ x - 5 = \frac{x}{5} \]

### Step 3: Solve for x

To eliminate the fraction, multiply both sides by 5:

\[ 5(x - 5) = x \]

Expanding the left side gives:

\[ 5x - 25 = x \]

Now, isolate x by moving x to the left side:

\[ 5x - x = 25 \]

This simplifies to:

\[ 4x = 25 \]

Dividing both sides by 4 gives:

\[ x = \frac{25}{4} \]

### Step 4: Solve the second equation `log(y - 6) = log(y) - log(6)`

Using the same property of logarithms:

\[ log(y - 6) = log\left(\frac{y}{6}\right) \]

### Step 5: Set the arguments of the logarithms equal to each other

Again, since the logarithm function is one-to-one, we set the arguments equal:

\[ y - 6 = \frac{y}{6} \]

### Step 6: Solve for y

Multiply both sides by 6 to eliminate the fraction:

\[ 6(y - 6) = y \]

Expanding the left side gives:

\[ 6y - 36 = y \]

Now, isolate y by moving y to the left side:

\[ 6y - y = 36 \]

This simplifies to:

\[ 5y = 36 \]

Dividing both sides by 5 gives:

\[ y = \frac{36}{5} \]

### Final Results

We have found:

\[ x = \frac{25}{4}, \quad y = \frac{36}{5} \]

### Conclusion

Now, we can analyze the results to determine which of the options given in the question is correct based on the values of x and y.

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