SOLUTION: Please help!!! Solve the equation. log(3+x)

SOLUTION: Please help!!! Solve the equation. log(3+x) - log(x-3) = log3

Question 85349: Please help!!!
Solve the equation.
log(3+x) - log(x-3) = log3
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Given:
.

.
Notice that the left side has the difference of two logs. By the rules of logs this can
be written as the log of the division of the two quantities that the log is operating
on, the numerator being from the positive log and the denominator being from the negative
log. Following this rule the equation becomes:
.

.
Note that for this equation to be true the quantities in the parentheses on both sides of
the equal sign must be equal. In other words, for the left side of this equation to be equal to
the right side, the log operator must be acting on equal quantities on both sides.
.
This being the case, you can write the new equation:
.

.
Get rid of the denominator by multiplying both sides of this equation by (x - 3). When
you do the equation becomes:
.

.
Multiplying out the right side leads to the equation:
.

.
Get rid of the 3x on the right side by subtracting 3x from both sides to get:
.

.
and then eliminate the 3 on the left side by subtracting 3 from both sides:
.

.
Finally, solve for x by dividing both sides by -2:
.

.
Check to ensure this is correct by returning to the original problem and substituting
6 for x:
.

.
and applying the rule for the difference of the two logs on the left side:
.

.
which simplifies to:
.

.
Since this is obviously true, we have a correct solution in x = +6.
.
Hope this helps you to understand the problem.
.

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