Solve the equation log2 x + log2 (x + 4) - log2 (x - 2) = 4

Answer 1

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.