Solve the equation:log_3(x²+x)-log_3(x²-x)=1

Answer 1

Using the logarithmic identity log_b(a) - log_b(c) = log_b(a/c), we can rewrite the equation as:

log_3((x^2 + x)/(x^2 - x)) = 1

Now, we can rewrite 1 as log_3(3^1), so:

log_3((x^2 + x)/(x^2 - x)) = log_3(3^1)

By equating the bases on both sides of the equation, we have:

(x^2 + x)/(x^2 - x) = 3^1

Simplifying the right side:

(x^2 + x)/(x^2 - x) = 3

To further simplify the equation, we can factor out an x on the left side:

x(x + 1)/(x(x - 1)) = 3

Now we can cancel out the common factors:

(x + 1)/(x - 1) = 3

To get rid of the fraction, we can cross-multiply:

(x + 1) = 3(x - 1)

Expanding the right side:

x + 1 = 3x - 3

Subtracting x from both sides:

1 = 2x - 3

Adding 3 to both sides:

4 = 2x

Dividing both sides by 2:

2 = x

Therefore, x = 2 is the solution to the equation.