Solve for x that satisfies the equation log (x+5)=log4-log(x+2)
To solve the equation log (x+5) = log4 - log(x+2), we can use the properties of logarithms.
First, we can simplify the right side of the equation using the property log a - log b = log(a/b):
log (x+5) = log(4/(x+2))
Now, we have the same logarithm on both sides of the equation, so we can remove the logarithm:
x + 5 = 4/(x+2)
Next, we can cross-multiply to eliminate the fraction:
(x+2)(x+5) = 4
Expanding the left side of the equation:
x^2 + 7x + 10 = 4
Rearranging the equation:
x^2 + 7x + 6 = 0
Now we can solve for x by factoring or using the quadratic formula:
(x+6)(x+1) = 0
This equation is satisfied if either x+6 = 0 or x+1 = 0:
If x+6 = 0, then x = -6
If x+1 = 0, then x = -1
Therefore, the values of x that satisfy the equation log (x+5) = log4 - log(x+2) are x = -6 and x = -1.
To solve the equation log (x+5) = log4 - log(x+2), we can use logarithmic properties to simplify it. Here are the steps:
Step 1: Combine the logs on the right side using the quotient rule:
log4 - log(x+2) = log(4/(x+2))
Step 2: Now we have log(x+5) = log(4/(x+2)). Since the two sides have the same base, we can set the arguments equal to each other. This gives us:
x+5 = 4/(x+2)
Step 3: Multiply both sides of the equation by (x+2) to eliminate the fraction:
(x+2)(x+5) = 4
Step 4: Expand the left side of the equation:
(x^2 + 2x + 5x + 10) = 4
Step 5: Combine like terms:
x^2 + 7x + 10 = 4
Step 6: Subtract 4 from both sides of the equation:
x^2 + 7x + 6 = 0
Step 7: Factor the quadratic equation:
(x+6)(x+1) = 0
Step 8: Set each factor equal to zero and solve for x:
x+6 = 0 or x+1 = 0
Step 9: Solve for x:
x = -6 or x = -1
So the solution to the equation log (x+5) = log4 - log(x+2) is x = -6 or x = -1.