You can put this solution on YOUR website!
Solving equations where the variable is in the argument of a logarithm, like this equation, usually starts with transforming the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)
With the "non-log" term of 1 in your equation, the first form will be easier to reach. We want just a single logarithm so we have to do is find a way to combine the two logarithms into one.
The terms on the left side of the equation are not like terms so we cannot just add to combine them. (Like logarithmic terms have logarithms of the same bases and same arguments.)
However there are two properties of logarithms which allow you to combine two into one:
These properties require logarithms of the same base with coefficients of 1. The two logs on the left meet these requirements. So we can use one of these properties to combine them. Since your logarithms have a "+" between them we will use the first property (which also has a "+".) Using the first property on your equation we get:
which simplifies to:
We now have the equation in the desired first form. With this form the next step is to rewrite the equation in exponential form. In general is equivalent to . Using this pattern (and the fact that the base of ln is "e") we get:
which simplifies to:
With the logarithm gone we can now solve for x. Since this is a quadratic equation we want one side to be zero. Subtracting 4 from each side we get:
Now we factor (or use the Quadratic Formula). This factors easily:
(x-4)(x+1) = 0
Form the Zero Product Property we know that one of these factors must be zero. So:
x-4 = 0 or x+1 = 0
Solving these we get:
x = 4 or x = -1
Answers to equations like yours must be checked. You must ensure the answers make all arguments to all logarithms positive. Any "solution" that makes an argument zero or negative must be rejected (because arguments to all logarithms must be positive). These rejected solutions can happen any time and when they do happen it is not a sign that a mistake was made! So even expert mathematicians must check their answers on these equations.
Use the original equation to check:
Checking x = 4:
We can already tell that both arguments will be positive when x = 4. So there is no reason to reject this solution. This is the required part of the check. The rest of the check is optional and will tell us if we made a mistake. You are welcome to finish the check.
Checking x = -1:
We can already tell that the arguments will be negative when x = -1. Since arguments of logarithms cannot be negative we will reject this solution. Notes:
If even just one argument had been negative (or zero), we would still reject this solution.
We rejected this solution not because x was negative but because the arguments became negative. In other equations, a negative could make the arguments positive. And on the other hand a positive value for x might make an argument negative. So you can't just accept positive solutions and reject zero or negative solutions. You have to see what happens to the arguments.
It is possible for all the the found solutions to work and it is also possible to reject all the solutions.
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