# How do you solve #ln (4x – 2) – ln 4 = - ln (x-2)#?

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To solve the equation ( \ln(4x - 2) - \ln(4) = -\ln(x - 2) ), we can use logarithmic properties to simplify it.

First, we'll apply the quotient rule of logarithms, which states that ( \ln(a) - \ln(b) = \ln(a/b) ). Using this rule, we can rewrite the left side of the equation as ( \ln\left(\frac{4x - 2}{4}\right) ).

So, the equation becomes ( \ln\left(\frac{4x - 2}{4}\right) = -\ln(x - 2) ).

Next, we'll use another property of logarithms, which states that ( \ln(a) = -\ln(b) ) if and only if ( a = 1/b ). Applying this property, we can rewrite the equation as ( \frac{4x - 2}{4} = \frac{1}{x - 2} ).

Now, we'll solve for ( x ):

( \frac{4x - 2}{4} = \frac{1}{x - 2} )

( 4x - 2 = 4 \cdot \frac{1}{x - 2} )

( 4x - 2 = \frac{4}{x - 2} )

Multiply both sides by ( x - 2 ) to clear the fraction:

( (4x - 2)(x - 2) = 4 )

Expand and simplify:

( 4x^2 - 8x - 2x + 4 = 4 )

( 4x^2 - 10x + 4 = 4 )

Subtract 4 from both sides:

( 4x^2 - 10x = 0 )

Factor out ( 2x ):

( 2x(2x - 5) = 0 )

Now, we have two possibilities:

- ( 2x = 0 ), which gives ( x = 0 ).
- ( 2x - 5 = 0 ), which gives ( x = \frac{5}{2} ).

However, we need to check these solutions to ensure they are valid. Since the natural logarithm is undefined for non-positive arguments, we must verify that both solutions satisfy the original equation.

Checking ( x = 0 ):

( \ln(4(0) - 2) - \ln(4) = \ln(-2) - \ln(4) ) is not valid.

Checking ( x = \frac{5}{2} ):

( \ln\left(4\left(\frac{5}{2}\right) - 2\right) - \ln(4) = \ln(8 - 2) - \ln(4) = \ln(6) - \ln(4) )

( -\ln\left(\frac{5}{2} - 2\right) = -\ln\left(\frac{5}{2} - 2\right) )

Both sides are equal, so ( x = \frac{5}{2} ) is a valid solution.

Therefore, the solution to the equation ( \ln(4x - 2) - \ln(4) = -\ln(x - 2) ) is ( x = \frac{5}{2} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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