The extraneous solutions that a logarithmic equation means the value of variables which is not in the domain of logarithm.

The logarithm equation have have infinitely many extraneous solutions may exist.

Concider a logarithm equation .

Apply product property of logarithm : .

Apply definition of logarithm : if and only if .

and

and .

Case 1 :

Che ck the logarithm at .

substitute in .

Apply power property of logarithm : .

.

Apply property of logarithm : .

.

The logarithm equation exist at .

Case 2 :

Che ck the logarithm at .

substitute in .

The domain of logarithm function is :  .

and not in the doain the domain of logarithm.

The logarithm equation that has extraneous solution at the value of is .

The logarithm equation have have infinitely many extraneous solutions may exist.