If log105 + log10 ( 5x+1 ) = log10 (x+8) + 1, then x is equal to ?

Solution:

Key steps:

• Use the product rule for logarithms: log10 5 + log10(5x+1) = log10[5(5x+1)].

• Rewrite 1 as a logarithm: 1 = log10 10, so log10(x+8) + 1 = log10(10(x+8)).

• Equate the arguments of the equal logarithms: 5(5x+1) = 10(x+8).

• Solve the linear equation: 25x + 5 = 10x + 80 ⇒ 15x = 75 ⇒ x = 5.

• Check domain conditions for logarithms: 5x+1 > 0 ⇒ x > -0.2, and x+8 > 0 ⇒ x > -8. The solution x = 5 satisfies both, so it is valid.

Therefore the correct value of x is 5.