SOLUTION: How do you solve: log(x-1)=log(x-2)-log(x+2)

Question 676821: How do you solve: log(x-1)=log(x-2)-log(x+2)
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 all the terms in your equation are logarithmic terms, the second, "all-log" form will be easiest to achieve. All we have to do is find a way to combine the two logs on the right side of the equation.

The two logs on the right are not like terms so we cannot simply subtract them. (Like logarithmic terms have the same bases and the same arguments.) But there is a property of logarithms, , which gives us another way to combine logarithmic terms which have a "-" between them. This property only requires the same bases and coefficients of 1. The logs on the right side fit both requirements:

We now have the desired form.

The next step with this form is based on some simple logic. The equation says that two base 10 logs are equal. This means that the exponent for 10 that results in x-1 is equal to (x-2)/x+2). In other words, x-1 and (x-2)/x+2) are equal to the same power of 10. The only way this can be true is if x-1 and (x-2)/x+2) are equal. So:

Now that the logs are gone, we can use "regular" algebra to solve. First we'll eliminate the fraction by multiplying both sides by (x+2):

which simplifies to:

This is a quadratic equation so we want one side to be zero. Subtracting x and adding 2 we get:

I hope its obvious that x = 0.

When solving logarithmic equations like this one you must check your answers. It is not optional. You must check to ensure that any possible solutions make all arguments to logarithms positive. Use the original equation to check:

Checking x = 0:

We can already see that two of the three logs will have negative arguments if x = 0. So x = 0 fails the check! We must reject x = 0 as a solution. (If even just one logarithm had had an argument that was negative or zero we would still reject the solution.) And since we rejected only "solution" we found, there is no solution to your equation. (IOW, the equation is impossible.)
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