Solve: log4 (7t + 2) = 2

log3 (x - 2) + log3 (x - 4) = log3 (2x^2 + 139) - log3 (3)

We now use the product and quotient rules of the logarithm to rewrite the equation as follows.

log3[ (x - 2) (x - 4) ] = log3[ (2x^2 + 139) / 3 ]

Which gives the algebraic equation

(x - 2) (x - 4) = (2x^2 + 139) / 3

Mutliply all terms by 3 and simplify

3 (x - 2) (x - 4) = (2x^2 + 139)

Solve the above quadratic equation to obtain

x = - 5 and x = 23

check:

1) x = - 5 cannot be a solution to the given equation as it would make the argument of the logarithmic functions on the right negative.

2) x = 23

Right Side of equation:

log3 (23 - 2) + log3 (23 - 4) = log3 (21*19) = log3 (399)

Left Side of equation:

log3 (2 (23) ^2 + 139) - 1 = log3 (1197) - log3 (3) = log3 (1197 / 3) = log3 (399)

conclusion: The solution to the above equation is x = 23