Question 826267
{{{log(4, (x)) + log(4, (12)) = log(4, (48))}}}
First we use a property of logarithms, {{{log(a, (p)) + log(a, (q)) = log(a, (p*q))}}}, to combine the two logs on the left:
{{{log(4, (x*12)) = log(4, (48))}}}
or
{{{log(4, (12x)) = log(4, (48))}}}
The equation now says that two base 4 logs are equal. The only way this can be true is if the arguments of both logs are equal, too. So:
4x = 12
Dividing by 4:
x = 3<br>
Checking is not optional when solving logarithmic equations like this. We must check to ensure that all bases and arguments are valid for the solution(s). (Valid bases are positive but not 1 and valid arguments are positive. Use the original equation to check:
{{{log(4, (x)) + log(4, (12)) = log(4, (48))}}}
Checking x = 3:
{{{log(4, (3)) + log(4, (12)) = log(4, (48))}}}
We can already see that all bases are 4 (valid) and all the arguments are positive (valid). So x = 3 checks out and is the solution.<br>
P.S. If x = 3 had made any base or any argument of any logarithm invalid, we would have to reject it, even if it is the only "solution" we found.