Log10 3+log10 (4x+1)=log10 (x+1)+1

Concept: For a fixed logarithm base, the product rule states logb(m) + logb(n) = logb(mn), and any real number k can be written as logb(bk) — in particular log10(10) = 1. When two logarithms with the same base are equal and both arguments are positive, the arguments themselves must be equal.

Applying this to the given equation:

1. Rewrite the constant 1 on the right-hand side as log10(10), so log10 3 + log10(4x+1) = log10(x+1) + 1 becomes log10 3 + log10(4x+1) = log10(x+1) + log10 10.

2. Apply the product rule to both sides: the left side combines to log10(3(4x+1)), and the right side combines to log10(10(x+1)).

3. Since the logarithms are equal (same base, positive arguments), equate the arguments: 3(4x+1) = 10(x+1), i.e. 12x + 3 = 10x + 10.

4. Solve the resulting linear equation for x: 12x + 3 = 10x + 10 gives 2x = 7, so x = 7/2.

Cross-check:

Substituting x = 7/2 back: 4x + 1 = 15 and x + 1 = 9/2, both positive, so every logarithm is defined. The left side becomes log10 3 + log10 15 = log10 45, and the right side becomes log10(9/2) + 1 = log10(9/2) + log10 10 = log10 45 — the two sides match, confirming x = 7/2 satisfies the equation.