Question

Select the correct answer. What is the solution to this equation? \( \ln(x^6) - \ln(2x-1) = 0 \)?

a. \( x = 1 \)
b. \( x = \frac{1}{2} \)
c. \( x = 0 \)
d. \( x = -1 \)

Answer 1

To solve the equation ln(x^6) - ln(2x-1) = 0, we need to rewrite it as x^6/(2x-1) = 1. None of the answer choices provided seem to be correct solutions to the equation.

Step-by-step explanation:

To solve the equation ln(x^6) - ln(2x-1) = 0, we can use the property of logarithms that states ln(a) - ln(b) = ln(a/b). Applying this property, we get ln(x^6/(2x-1)) = 0. Since the natural logarithm of 1 is 0, we can conclude that x^6/(2x-1) = 1.

Multiplying both sides of the equation by (2x-1), we get x^6 = 2x-1. This is a polynomial equation of degree 6. To solve it, we need to either factorize it or use numerical methods. Since it is not factorizable easily, we can solve it numerically using graphing or root-finding methods.

None of the answer choices provided seem to be correct solutions to the equation. Therefore, none of the options a, b, c, or d are correct.