SOLUTION: f(x) = log2(x - 3) and g(x) = log2(3x + 17). Solve f(x) + g(x) = 6.

Question 229527: f(x) = log2(x - 3) and g(x) = log2(3x + 17).
Solve f(x) + g(x) = 6.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Substituting f(x) and g(x) we get:

To solve and equation like this, with the variable in the argument of a logarithm, we need to transform the equation into one of the following the forms:
log(expression with x) = other expression
log(expression with x) = log(other expression with x)

With the "non-log" term of 6, the second form will be difficult. So we will aim for the first form. All we need to do, somehow, is to combine the to logarithms into one. And luckily there is a property of logarithms which allows us to do exactly that: . This property allows us to take the sum of two logarithms of the same base and rewrite them as a single logarithm of the product of the original arguments. So:

can be rewritten as:

and we have the required form. To solve this we rewrite it in exponential form:

The logarithms are now gone and we solve this as a "normal" equation. We'll start by simplifying:

Since this is a quadratic equation, we'll get one side equal to zero:

We can solve this by factoring or by using the quadratic formula. This will factor:

According to the Zero Product Property, we know that this product can be zero only if one of the factors is zero:
or
Solving these we get:
or

With logarithm equations we need to check the answers we get to make sure they do not make an argument negative or zero (aka non-positive). (There's no such thing as a log of a negative or of zero. After all, how do you raise a positive number to a power and get a negative or zero result? Answer: You can't.)

Checking x = -23/3:

We have to reject x = -23/3 because, as you can see, it makes the arguments negative. (Even if it made only one argument negative we would have to reject it.)

Checking x = 5:

This is going to work because both arguments are going to be positive. Finishing the check:

Check!
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