Question 413132: Can you please help me show the steps on how to answer this equation?
log(base)2 of (x-3) + log(base)2 of (x+1)= log(base)2 of (6x-18) Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
Solving equations where the variable is in the argument of a logarithm usually starts with transforming the equation into one of the following forms:
log(expression) = expression
or
log(expression) = log(expression)
Since your equation is made up of entirely with logarithmic terms, the second, "all-log" form will be easier to achieve. All we have to do is find a way to combine the two logarithms on the left into a single logarithm.
The terms on the left are not like terms so we cannot just add them together. (Like logarithmic terms have the same bases and same arguments.)
Fortunately there is another way to combine logarithmic terms. Two properties of logarithms:
These properties require the same base and coefficients of 1. Your logarithms meet both requirements so we can use the first property (because of the "+" between the logarithms) to combine them:
which simplifies to:
We finally have the second form. With this form the next step is based on some simple logic. The equation says two base 10 logarithms are equal. The only way that two logarithms of tha same base can be equal is if the arguments are equal. So:
We now have a quadratic equation to solve. So we want one side to be zero. Subtracting 6x and add 18 we get:
Now we factor (or use the Quadratic Formula). This factors easily:
(x-3)(x-5) = 0
From the Zero Product Property we know that one of these factors must be zero. So:
x-3 = 0 or x-5 = 0
Solving each of these we get:
x = 3 or x = 5
Solving logarithmic equations like your requires that answers be checked to make sure that all arguments of logarithms are positive. Use the original equation to check:
Checking x = 3:
We can see already that the first and last arguments are going to be zero when x = 3. Arguments of logarithms cannot be zero (or negative) so we must reject this solution. (If only one argument had been "bad" (zero or negative) we would still reject this solution.) Rejecting this solution does not mean we made a mistake. It is just something that can happen any time you solve these kinds of equations. This is why the check is required, even for expert mathematicians!
Checking x = 5:
We can see already all arguments will be positive when x = 5. So there is no reason to reject this solution. The reset of the check will just tell us if we made a mistake. You are welcome to finish the check.
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