You can put this solution on YOUR website! log(x) + log(x-14) = log(2x)
Solving equations like this, where the variable is in the argument of a logarithm, usually start with transforming the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)
Since your equation has nothing but logarithmic terms, the second form will be easier to achieve. All we need to do to achieve the second form is find a way to combine the two terms on the left into one.
The two 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. Your terms have the same base, 10, but different arguments, x and x-14.)
Fortunately there are a couple of properties of logarithms which allow us to combine two logarithmic terms into one:
These properties require the same bases and coefficients of 1. (The arguments can be anything.) Your logarithms fit both requirements. Since your logarithms have a "+" between them we will use the first property:
log(x*(x-14)) = log(2x)
which simplifies to:
We now have the second form. With this form the next step uses some simple logic. This equation says that two base 10 logarithms are equal. The only way two logarithms of the same base can be equal is if the arguments are equal too. So:
This equation we can solve. It is quadratic so we want one side to be zero. Subtracting 2x from each side we get:
Now we factor (or use the Quadratic Formula). This factors very easily:
x(x-16) = 0
From the Zero Product Property we know that this (or any) product can be zero only if one (or more) of the factors is zero. So:
x = 0 or x-16 = 0
Solving the second equation we get:
x = 0 or x = 16
When solving these equations you must check your answers. You must ensure that each "solution" makes all the arguments of logarithms positive. Any "solution" that makes an argument zero or negative must be rejected sicne arguments of logarithms must be positive. These solutions that get rejected can happen even if no mistakes were made! So even expertes in Math must check their solutions.
Use the original equation when checking:
log(x) + log(x-14) = log(2x)
Checking x = 0:
log(0) + log(0-14) = log(2(0))
We can already see that we get two zero arguments and one negative one. So we must reject this solution. (If even only one argument was zero or negative we would reject this solution.) Rejecting this solution does not mean we made a mistake earlier!
Checking x = 16:
log(16) + log(16-14) = log(2(16))
We can already see that all the arguments will be positive when x = 16. So there is no reason to reject this solution. The remainder of the check will tell us if we made a mistake. You are welcome to finish the check.
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