If $$\log x + \log y = \log (x + y)$$, find the relationship between $$x$$ and $$y$$.

Question

Accepted Answer

To solve the equation $$\log x + \log y = \log (x + y)$$, we use the properties of logarithms step by step:

**Step 1: Apply the logarithmic property**  
Using the property of logarithms, $$\log a + \log b = \log (ab)$$, the left-hand side becomes:  
$$
\log(xy) = \log(x + y)
$$

**Step 2: Remove the logarithm**  
Since the logarithm function is one-to-one, we equate the arguments:  
$$
xy = x + y
$$

**Step 3: Rearrange the equation**  
Rewriting the equation gives:  
$$
xy - x - y = 0
$$  
Factoring out common terms:  
$$
y(x - 1) = x
$$

**Step 4: Solve for $$y$$**  
Dividing both sides by $$(x - 1)$$ (assuming $$x 
eq 1$$):  
$$
y = \frac{x}{x - 1}
$$

Thus, the relationship between $$x$$ and $$y$$ is **$$y = \frac{x}{x - 1}$$**.

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