Solve the equation:log_3(x²+x)-log_3(x²-x)=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.
To solve the equation:
log_3(x²+x) - log_3(x²-x) = 1
We can use the properties of logarithms to simplify the equation. One property states that the difference of logarithms with the same base is equal to the logarithm of the quotient of their arguments:
log_3((x²+x)/(x²-x)) = 1
Now, let's simplify the expression inside the logarithm:
(x²+x)/(x²-x)
To simplify this expression, we can factor out an x from the numerator and the denominator:
x(x+1)/x(x-1)
Now, we can cancel out the common factor of x:
(x+1)/(x-1)
Now, let's rewrite the equation:
log_3((x+1)/(x-1)) = 1
To get rid of the logarithm, we can convert the equation into exponential form:
3^1 = (x+1)/(x-1)
Simplifying the left side:
3 = (x+1)/(x-1)
Now, let's solve for x. Multiply both sides of the equation by (x-1):
3(x-1) = x+1
Distribute:
3x - 3 = x + 1
Subtract x from both sides:
2x - 3 = 1
Add 3 to both sides:
2x = 4
Divide both sides by 2:
x = 2
Therefore, the solution to the equation is x = 2.