If log5 + log( 5x+1 ) = log(x+5) + 1, then x is equal to ?

Concept: For logarithms of the same base, the product rule gives log b(m) + log b(n) = log b(mn), and any constant k can be rewritten as k = log b(b^k) (in base 10, 1 = log 10). Solving a logarithmic equation means collapsing both sides into a single logarithm of the same base and then equating the arguments (after checking the arguments stay positive).

Application:

1. Combine the left side with the product rule: log5 + log(5x+1) = log[5(5x+1)] = log(25x+5).

2. Rewrite the constant on the right as a log10 and combine: log(x+5) + 1 = log(x+5) + log10 = log[10(x+5)] = log(10x+50).

3. Since both sides are now log of the same base, their arguments must be equal: 25x + 5 = 10x + 50.

4. Solve the linear equation: 15x = 45, so x = 3.

5. Check the domain: at x = 3, 5x+1 = 16 > 0 and x+5 = 8 > 0, so both logarithms are defined.

Cross-check:

Substituting x = 3 directly into the original equation: left side = log5 + log16 = log80; right side = log8 + 1 = log8 + log10 = log80. Both sides equal log80, confirming x = 3 satisfies the equation.