If 2log10(x + 1) = log10(7x + 1), then find the non-zero value of 'x'?

Solution:

1. Start with the given equation: 2·log10(x + 1) = log10(7x + 1).

2. Use the logarithm power rule: 2·log10(x + 1) = log10((x + 1)2). So the equation becomes log10((x + 1)2) = log10(7x + 1).

3. Since log10 is one-to-one on positive arguments, equate the insides: (x + 1)2 = 7x + 1.

4. Expand and simplify: x2 + 2x + 1 = 7x + 1 ⇒ x2 − 5x = 0 ⇒ x(x − 5) = 0.

5. So x = 0 or x = 5. Check domain: log arguments require x + 1 > 0 and 7x + 1 > 0, which both hold for these values. The question asks for the non-zero value, so choose x = 5.

Answer: 5