To solve this logarithmic equation, we can use the properties of logarithms to combine the two logarithms on the left side of the equation.
First, recall that log a + log b = log ab.
Using this property, we can rewrite the left side of the equation as a single logarithm:
log3((5-x)(-1-x))
Next, simplify the expression within the logarithm:
(5-x)(-1-x) = -5 + x + x - x^2 = -5 - x - x^2
Therefore, the left side of the equation simplifies to:
log3(-5 - x - x^2)
Now, set this equal to -3:
log3(-5 - x - x^2) = -3
To solve for x, we need to exponentiate both sides with base 3:
3^log3(-5 - x - x^2) = 3^(-3)
This simplifies to:
-5 - x - x^2 = 1/27
Rearranging the equation, we get a quadratic equation:
x^2 + x + 5 + 1/27 = 0
Solving this quadratic equation will give us the possible values of x. We could use the quadratic formula or factoring to find the solutions for x.
To solve this logarithmic equation, we can use the properties of logarithms to combine the two logarithms on the left side of the equation.
First, recall that log a + log b = log ab.
Using this property, we can rewrite the left side of the equation as a single logarithm:
log3((5-x)(-1-x))
Next, simplify the expression within the logarithm:
(5-x)(-1-x) = -5 + x + x - x^2 = -5 - x - x^2
Therefore, the left side of the equation simplifies to:
log3(-5 - x - x^2)
Now, set this equal to -3:
log3(-5 - x - x^2) = -3
To solve for x, we need to exponentiate both sides with base 3:
3^log3(-5 - x - x^2) = 3^(-3)
This simplifies to:
-5 - x - x^2 = 1/27
Rearranging the equation, we get a quadratic equation:
x^2 + x + 5 + 1/27 = 0
Solving this quadratic equation will give us the possible values of x. We could use the quadratic formula or factoring to find the solutions for x.