Question 419692: ln(x-7) - ln (2)= 3
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! ln(x-7) - ln(2) = 3
Solving equations where the variable is in the argument of alogarithm, like this equation, usually starts with transforming the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = (other-expression)
With the "non-log" term of 2, the second form, which is "all-log" will be more difficult to achieve. So we will aim for the first form. All we need to do to achieve the first form is find a way to combine the two logarithms into one.
The two logs are not like terms so we cannot just subtract them. (Like logarithmic terms have the same bases and the same arguments. Your terms have the same base, e, but the arguments, (x-7) and 2, are different.)
Fortunately there is another way to combine logarithms. Two properties of logarithms:These properties require logarithms of the same base with coefficients of 1. Your logarithms meet both requirements. So we can use the second property (because just like your logarithms there is a "-" between them):
We now have the first form. With the first form the next step is to rewrite the equation in exponential form. In general,  is equivalent to  . Using this pattern with your equation we get:
Now that the variable is out of the logarithm we can solve for it. Multiplying both sides by 2 we get:
Adding 7 to each side we get:
With these equations you must check your answer(s). You must make sure your solution(s) make all arguments of all logarithms positive. Any "solution" that makes an argument of a logarithm zero or negative must be rejected. These rejected solutions, when they happen, do not mean that some mistake was made.
Use the original equation to check:
ln(x-7) - ln(2) = 3
Checking :
The second argument, 2, is clearly positive. For the first argument...- "e" is a positive number.
- Since e is positive
 will be positive. - Since is positive,
 will be positive. - Since is positive,
 will be greater than 7. - Since is greater than 7,
 will be positive.
So both arguments are positive. There is no reason to reject this solution. (This is the required part of the check. The remainder of the check is optional and will tell us if we made a mistake. You are welcome to finish the check if you like.
So is the exact expression for the solution to your equation. (If you want/need a decimal approximation then replace e with a decimal approximation for it and simplify.)
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