Question

Solve the equation and state the extraneous solutic log₃(x)+log₃(x-6)=3 The real solution is x

Answer 1

To solve the equation log₃(x) + log₃(x - 6) = 3, combine the logs, expand and simplify, then solve for x. The extraneous solution is rejected, leaving the real solution x = 9.

Step-by-step explanation:

To solve the equation log₃(x) + log₃(x - 6) = 3, we can combine the logs using logarithm properties.

1. Start by using the product rule of logarithms:

log₃(x) + log₃(x - 6) = log₃(x(x - 6))

1. Apply the logarithmic form of the equation:

x(x - 6) = 3³

1. Expand and simplify the equation:

x² - 6x = 27

1. Move all terms to one side to set the equation equal to zero:

x² - 6x - 27 = 0

1. Factor the quadratic equation or use the quadratic formula:

(x - 9)(x + 3) = 0
x = 9 or x = -3

However, x = -3 is an extraneous solution because it results in the log of a negative number, which is not defined. Therefore, the real solution to the equation is x = 9.