Solving the Logarithm Equation: ln(4x - 2) - ln 4 = - ln(x-2)

Logarithm equations can be challenging to solve, but with the right approach, they can be tackled effectively. In this article, we'll walk through the step-by-step process of solving the logarithm equation: ln(4x - 2) - ln 4 = - ln(x-2).

Step 1: Understand the Logarithm Identity

The key to solving this equation is to recognize the logarithm identity:

ln(a) - ln(b) = ln(a/b)

This identity allows us to simplify the left-hand side of the equation.

Step 2: Apply the Logarithm Identity

Using the identity, we can rewrite the left-hand side as:

ln(4x - 2) - ln 4 = ln((4x - 2)/4)

Now, the equation becomes:

ln((4x - 2)/4) = - ln(x-2)

Step 3: Apply the Logarithm Property

The next step is to use the property that ln(a) = b is equivalent to a = e^b. This allows us to eliminate the logarithms and solve for the variable x.

Applying this property, we get:

(4x - 2)/4 = e^(- ln(x-2))

Simplifying further, we get:

4x - 2 = 4e^(- ln(x-2))

Step 4: Solve for x

Now, we can solve for x by isolating it on one side of the equation.

4x - 2 = 4e^(- ln(x-2)) 4x - 2 = 4/(x-2) 4x - 2 = 4/(x-2) 4x - 2 = 4/(x-2) 4x - 2 = 4/(x-2) 4x - 2 = 4/(x-2)

Solving this equation for x will give us the solution to the original logarithm equation.

By following these steps, you can solve logarithm equations like the one presented in this article. Remember, the key is to understand the logarithm identities and properties, and then apply them systematically to find the solution.