Question

What is the solution to the following equation log6x + log6 (x - 9) = 2

A. X = 12.
B. X= 22.5.
C. X = -3 or x = 12.
D. X= 3 or x = -12

Answer 1

The equation \log6(x) + log6(x - 9) = 2 has a solution of x = 12 after using logarithm properties to combine terms, exponentiate to remove the logarithm, and solving the resulting quadratic equation.

Step-by-step explanation:

To find the solution to the equation log6(x) + log6(x - 9) = 2, we can use the properties of logarithms to combine the two logarithmic expressions on the left side of the equation.

Using the product rule of logarithms, which states that logb(M) + logb(N) = logb(MN) when M and N are positive, we obtain:

log6(x(x - 9)) = 2

Exponentiating both sides with base 6 to remove the logarithm, we get:

x(x - 9) = 62

Simplify and solve for x:

x2 - 9x - 36 = 0

Factoring the quadratic equation, we find the solutions for x:

x = 12 or x = -3

However, since a logarithm is only defined for positive arguments and x cannot be negative, we discard x = -3. The only valid solution is x = 12 (Option A).