SOLUTION: Please help me solve this problem: log1/3(x^2+x)-log1/3(x^2-x)=-1. Do you have to divide the two logs because there is subtraction, or is there another way?

Question 708299: Please help me solve this problem: log1/3(x^2+x)-log1/3(x^2-x)=-1. Do you have to divide the two logs because there is subtraction, or is there another way?
Found 2 solutions by jsmallt9, nerdybill:
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

Solving equations like this usually starts with transforming the equation into one of the following forms:
log(expression) = number
or
log(expression) = log(expression)

With the "non-log" term of -1 on the right side it will be harder to transform your equation into the second, "all-log" form. So we will aim for the first form. This will require that we find a way to combine the two logs into one.

If they were like terms then we could just subtract them. But like logarithmic terms have the same bases and same arguments. Our logs have the same bases, 1/3, but the arguments are different. So we cannot subtract them.

Another way to combine logs is to one of the following properties:
These properties require the same bases and coefficients of 1. Our logs meet these requirements. We will use the second property because its logs, like ours, have a "-" between them:

[Note: This property changes the subtraction of two logs not into a division of the logs as you asked but into the log of the division of the arguments. This is a subtle but important difference!]

Our equation is now in the first form. The next step with this form is to rewrite the equation in exponential form. In general, is equivalent to . Using this pattern on our equation we get:

Since -1 as an exponent means reciprocal the left side simplifies to:

Now that the x's are out of logarithms we can solve the equation. First let's get rid of the fraction by multiplying each side by :

which simplifies to:

This is a quadratic equation so we want one side to be zero. Subtracting the entire right side from both sides we get:

Now we factor (or use the Quadratic Formula). This factors easily:

From the Zero Product Property:
2x = 0 or x-2 = 0
Solving these we get:
x = 0 or x = 2

Last of all we check. This is not optional when solving logarithmic equations like or this when you multiply both sides of an equation by an expression that might be zero. We have done both! We must at least check that the solutions create valid arguments and bases to the logarithms. Use the original equation to check:

Checking x = 0:

We can already see that both arguments are going to be zero if x = 0. Zero is not a valid argument for logarithms. (Valid arguments are positive.) So we must reject this solution.
Checking x = 0:

Simplifying...

Using the property to combine the logs:

Check! (If you cannot see that this is a true statement then rewrite this in exponential form.)

So the only solution to the equation is: x = 2
Answer by nerdybill(7384) (Show Source): You can put this solution on YOUR website!
Please help me solve this problem: log1/3(x^2+x)-log1/3(x^2-x)=-1. Do you have to divide the two logs because there is subtraction, or is there another way? yes, that's the correct approach.
.
I'm going to assume you meant log base (1/3):

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