log x (x+2)=2
is that log(x(x+2))=2 or is it (log(x)) times (X+2)=2?
(logx)(x+2)=2
more than likely you want
logx (x+2) = 2
then
x^2 = x+2
x^2-x-2 = 0
(x-2)(x+1) = 0
x = 2 or x = -1
but the base of any log has to be positive so,
x = 2
To solve the equation logₓ(x+2) = 2, you need to isolate x by using algebraic manipulation. Here's the step-by-step process:
Step 1: Rewrite the equation in exponential form.
In logarithmic form, logₓ(a) = b can be written as xᵇ = a. Applying this form to the equation logₓ(x+2) = 2, we get the equation x² = (x+2).
Step 2: Simplify and bring all terms to one side.
Expand x² to x * x and then subtract (x+2) from both sides. This gives us the quadratic equation x² - x - 2 = 0.
Step 3: Factorize the quadratic equation.
Factorizing the quadratic equation gives us (x-2)(x+1) = 0.
Step 4: Solve for x.
Set each factor to zero and solve for x independently:
x - 2 = 0 --> x = 2
x + 1 = 0 --> x = -1
So, the solution to the equation logₓ(x+2) = 2 is x = 2 or x = -1. Remember to check these solutions by substituting them back into the original equation to ensure they satisfy the logarithmic condition.