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To solve logarithmic equations where the variable is in one (or more) arguments of logarithms, we generally start by transforming the equaiton into one of the following forms:
With the non-logarithm term, the 3, the second form will not be easy to achieve. So we will aim for the first form. The first form has a single logarithm. Our equation has two. Somehow we need to combine them. Fortunately we have a property of logarithms which will allow us to combine our two logarithms into one:
Using this on our equation we get:
Now we just get that logarithm by itself by adding 3 to both sides of the equation:
We have finally achieved the first form. With this form we solve it by rewriting the equation in exponential form. To do this we need to know that is equivalent to . Using this on our equation we get:
which simplifies to:
This an equation we can solve. It is quadratic so we will get one side equal to zero by subtracting 8 from each side:
Now we either factor and use the Zero Product Property or use the Quadratic Formula. This factors fairly easily:
According to the Zero Product Property this (or any) product can be zero only if one (or more) factors is zero. So the solution will be: or
Solving each of these we get: or or
With logarithmic equations we should check our answers, not so much to see if they work but to see if they make an argument to a logarithm zero or negative. Any such answer must be rejected because we cannot allow arguments of logarithms to be zero or negative.
Always use the original equation to check.
Checking x = 2/3:
At this point we can see that neither argument will end up negative or zero. So it looks like 3/2 will work. You are welcome to finish the check.
Checking x = -4:
At this point we can see that both arguments will become negative. So we can stop here and reject x = -4 as a solution. (Even if only one of the two arguments had become negative or zero, we would still reject the solution.)
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