Question

Solve the logarithmic equation for x: log₁6(x−4)−1og₁6(x²)=1.

a) 0
b) 2
c) 4
d) 8

Answer 1

To solve the logarithmic equation log₁6(x−4)−1og₁6(x²)=1, we need to simplify the equation using logarithmic properties. After simplifying the equation, we find that the solution is x = 4. Therefore, the correct option is c) 4.

Step-by-step explanation:

To solve the logarithmic equation log₁6(x−4)−1og₁6(x²)=1, we need to use logarithmic properties to simplify the equation.

1. First, simplify the left side of the equation by applying the power rule of logarithms. We know that loga(xy) = y * loga(x). So, the equation becomes: log₁6(x-4) - 2log₁6(x) = 1.
2. Next, we can combine the logarithms using the subtraction property of logarithms, which states that loga(b) - loga(c) = loga(b/c). Applying this property, the equation becomes: log₁6((x-4)/x²) = 1.
3. To eliminate the logarithm, we can rewrite the equation in exponential form. For any logarithmic equation of the form loga(x) = y, we know that ay = x. Using this property, we have: ₁61 = (x-4)/x². Solving for x, we get: x = 4.

Therefore, the solution to the equation is x = 4. So, the correct option is c) 4.