You can put this solution on YOUR website! ln(2) + ln(x-6) = ln(x+2)
For equations where the variable is in the argument (or base) of a logarithm, you often start by transforming the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)
Since all the terms of your equation are logarithms, the first form will be more difficult to achieve. So we will aim for the second form. The right side is already a base e logarithm. We just need the left side to be a base e logarithm, too. So somehow we need to combine the two base e logarithms on the left side into one base e logarithm.
Fortunately there is a property of logarithms, , which allows us to combine two logarithms of the same base (with coefficients of 1 and a plus sign between them) into a single logarithm of that base. Your two logarithms fit the pattern for this property so we can use it to combine them into one:
ln(2*(x-6)) = ln(x+2)
We have achieved the second form. With the second form we use some basic logic. ln(2*(x-6)) represents the exponent for e that results in 2*(x-6). And ln(x+2) represents the exponent for e that results in x+2. The equation tells us that these two exponents are equal. And if equal exponents are place on e, then the results should be the same, too. So:
2*(x-6) = x+2
This is a very easy equation to solve. First we simplify:
2x - 12 = x + 2
Next we'll subtract x from each side:
x - 12 = 2
Add 12 to both sides:
x = 14
When solving logarithmic equations it is important, not just a good idea, to check your answers. You must make sure that your solution(s) make all arguments (and bases) of logarithms remain positive. Any solution that makes an argument (or base) zero or negative must be rejected. And these rejected solutions can happen even when no mistakes were made in finding them! This is why it is important to check
Always use the original equation to check:
ln(2) + ln(x-6) = ln(x+2)
Checking x = 14:
ln(2) + ln((14)-6) = ln((14)+2)
It should be easy to see that all three arguments will be positive when x = 14. So there is no reason to reject it. (If an argument had been zero or negative we would reject our only solution! This would mean there are no solutions to the equation.)
The rest of the check will tell us if we did make a mistake. You are welcome to complete the check.
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