Solve for the equation:log_4 x+log_4(x+6)=2
Using the properties of logarithms, we can rewrite the equation as:
log_4(x(x+6)) = 2
Using the logarithmic property log_a(b) = c can be rewritten as a^c = b:
4^2 = x(x+6)
16 = x^2 + 6x
Moving all terms to one side, we get:
x^2 + 6x - 16 = 0
We can solve this quadratic equation by factoring:
(x + 8)(x - 2) = 0
Setting each factor equal to zero:
x + 8 = 0 or x - 2 = 0
x = -8 or x = 2
However, we cannot take the logarithm of a negative number, so the only valid solution is x = 2.
To solve the equation log_4(x) + log_4(x+6) = 2, we can use the properties of logarithms to simplify the equation and find the value of x.
Step 1: Simplify the equation using the logarithmic property log(a) + log(b) = log(ab):
log_4(x(x+6)) = 2
Step 2: Rewrite the equation in exponential form:
4^2 = x(x+6)
Step 3: Simplify the exponential equation:
16 = x(x+6)
Step 4: Expand the equation:
16 = x^2 + 6x
Step 5: Rearrange the equation and set it equal to zero:
x^2 + 6x - 16 = 0
Step 6: Factor the quadratic equation:
(x + 8)(x - 2) = 0
Step 7: Set each factor equal to zero and solve for x:
x + 8 = 0 or x - 2 = 0
x = -8 or x = 2
Step 8: Check the solutions by substituting them back into the original equation:
Checking x = -8:
log_4(-8) + log_4(-8+6) = 2
The logarithm of a negative number is undefined, so x = -8 is not a valid solution.
Checking x = 2:
log_4(2) + log_4(2+6) = 2
Simplifying:
1 + log_4(8) = 2
log_4(8) = 2 - 1
log_4(8) = 1
Since 4^1 = 8, the equation is satisfied when x = 2.
Therefore, the solution to the equation log_4(x) + log_4(x+6) = 2 is x = 2.