If log10 a + log10 b = log10 (a + b), then the values of a and b are given by…

The logarithm product rule states that for a valid base and positive arguments, log base b of x plus log base b of y equals log base b of (xy) - the sum of two logarithms with the same base equals the logarithm of the product of their arguments. Because log base b is a one-to-one function on its domain, log base b of M equal to log base b of N implies M = N, for M, N > 0.

Applying this to the given equation, with base 10 throughout:

1. log base 10 of a plus log base 10 of b equals log base 10 of (ab), by the product rule.

2. The given equation log base 10 of a plus log base 10 of b equals log base 10 of (a + b) becomes log base 10 of (ab) equals log base 10 of (a + b).

3. Since log base 10 is one-to-one on positive reals, the arguments must be equal: ab = a + b.

4. Collect the terms containing a on one side: ab minus a equals b.

5. Factor out a: a times (b minus 1) equals b.

6. Divide both sides by (b minus 1), valid for b not equal to 1: a equals b divided by (b minus 1).

This requires a > 0 and b > 0 for the original logarithms to be defined, and b not equal to 1 so the division is valid.

Independent check: take b = 3, so a = 3 divided by (3 minus 1) = 1.5. Then log base 10 of a plus log base 10 of b is approximately 0.176 + 0.477 = 0.653, and log base 10 of (a + b) is log base 10 of 4.5, approximately 0.653 - the two sides agree, confirming a equals b divided by (b minus 1).