If log(a + b) = log a + log b, find a in terms of b.

Answer

To solve the equation \( \log(a + b) = \log a + \log b \) and find \( a \) in terms of \( b \), we can follow these steps:

### Step 1: Use the property of logarithms
We know that the property of logarithms states that:
\[ \log a + \log b = \log(ab) \]
So, we can rewrite the right-hand side of the equation:
\[ \log(a + b) = \log(ab) \]

### Step 2: Set the arguments equal
Since the logarithmic function is one-to-one, we can set the arguments equal to each other:
\[ a + b = ab \]

### Step 3: Rearrange the equation
To isolate \( a \), we can rearrange the equation:
\[ ab - a - b = 0 \]

### Step 4: Factor the equation
We can factor the left-hand side:
\[ a(b - 1) - b = 0 \]
This can be rearranged to:
\[ a(b - 1) = b \]

### Step 5: Solve for \( a \)
Now, we can solve for \( a \) by dividing both sides by \( (b - 1) \):
\[ a = \frac{b}{b - 1} \]

### Final Answer
Thus, we have found \( a \) in terms of \( b \):
\[ a = \frac{b}{b - 1} \]

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