Answer:
x = 3 + sqrt(e^3 + 9)
Explanation:
Solve for x over the real numbers:
log(x) + log(x - 6) = 3
Hint: | Combine logarithms.
log(x - 6) + log(x) = log(x (x - 6)):
log(x (x - 6)) = 3
Hint: | Eliminate the logarithm from the left hand side.
Cancel logarithms by taking exp of both sides:
x (x - 6) = e^3
Hint: | Write the quadratic polynomial on the left hand side in standard form.
Expand out terms of the left hand side:
x^2 - 6 x = e^3
Hint: | Take one half of the coefficient of x and square it, then add it to both sides.
Add 9 to both sides:
x^2 - 6 x + 9 = e^3 + 9
Hint: | Factor the left hand side.
Write the left hand side as a square:
(x - 3)^2 = e^3 + 9
Hint: | Eliminate the exponent on the left hand side.
Take the square root of both sides:
x - 3 = sqrt(e^3 + 9) or x - 3 = -sqrt(e^3 + 9)
Hint: | Look at the first equation: Solve for x.
Add 3 to both sides:
x = 3 + sqrt(e^3 + 9) or x - 3 = -sqrt(e^3 + 9)
Hint: | Look at the second equation: Solve for x.
Add 3 to both sides:
x = 3 + sqrt(e^3 + 9) or x = 3 - sqrt(e^3 + 9)
Hint: | Now test that these solutions are correct by substituting into the original equation.
Check the solution x = 3 - sqrt(e^3 + 9).
log(x) + log(x - 6) ⇒ log((3 - sqrt(e^3 + 9)) - 6) + log(3 - sqrt(e^3 + 9)) = 2 i π + 3 ≈ 3 + 6.28319 i:
So this solution is incorrect
Hint: | Check the solution x = 3 + sqrt(e^3 + 9).
log(x) + log(x - 6) ⇒ log(3 + sqrt(e^3 + 9)) + log((3 + sqrt(e^3 + 9)) - 6) = 3:
So this solution is correct
Hint: | Gather any correct solutions.
The solution is:
Answer: x = 3 + sqrt(e^3 + 9)