SOLUTION: solve for x log2 x + log2 (3x+10)-3=0

Question 245374: solve for x
log2 x + log2 (3x+10)-3=0
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

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:

log(expression-with-vairable) = other-expression
log(expression-with-variable) = log(other-expression)

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.)

So there is just the one solution: x = 2/3.
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