Solve the equation log2 x + log2 (x + 4) - log2 (x - 2) = 4
To solve the equation log2 x + log2 (x + 4) - log2 (x - 2) = 4, we can use the properties of logarithms to simplify the equation.
First, we can combine the logarithms using the product and quotient rules:
log2 [(x(x + 4))/(x - 2)] = 4
Next, we can convert the logarithmic equation into exponential form:
2^4 = (x(x + 4))/(x - 2)
16 = (x(x + 4))/(x - 2)
Now, we can cross-multiply:
16(x - 2) = x(x + 4)
16x - 32 = x^2 + 4x
Rearranging the equation to form a quadratic equation:
x^2 + 4x - 16x + 32 = 0
x^2 - 12x + 32 = 0
Now, we can solve this quadratic equation by factoring or by using the quadratic formula.
By factoring:
(x - 4)(x - 8) = 0
This gives two possible solutions:
x - 4 = 0
x = 4
x - 8 = 0
x = 8
Therefore, the solutions to the equation log2 x + log2 (x + 4) - log2 (x - 2) = 4 are x = 4 and x = 8.
To solve the equation log2 x + log2 (x + 4) - log2 (x - 2) = 4, we can use logarithmic properties to simplify the equation.
Step 1: Combine the logarithms using the following properties:
log a + log b = log (a * b)
log a - log b = log (a / b)
By applying these properties, we get:
log2 (x * (x + 4) / (x - 2)) = 4
Step 2: Rewrite the equation in exponential form:
2^4 = x * (x + 4) / (x - 2)
Simplifying further,
16 = x * (x + 4) / (x - 2)
Step 3: Multiply both sides of the equation by (x - 2) to eliminate the fraction:
16(x - 2) = x * (x + 4)
Expanding both sides,
16x - 32 = x^2 + 4x
Step 4: Rearrange the equation to have all the terms on one side:
x^2 + 4x - 16x + 32 = 0
Simplifying,
x^2 - 12x + 32 = 0
Step 5: Factorize the quadratic equation:
(x - 8)(x - 4) = 0
Setting each factor equal to zero,
x - 8 = 0 or x - 4 = 0
Solving for x,
x = 8 or x = 4
Therefore, the solutions to the equation log2 x + log2 (x + 4) - log2 (x - 2) = 4 are x = 8 and x = 4.