Solve ln (2x-3)= ln x-2 ln 2

Question

Solve the equation

x = \frac{12}{7}

Alternative Form

x = 1. \dot{7} 1428 \dot{5}

Evaluate

ln (2 x - 3) = ln (x) - 2 ln (2)

Find the domain

More Steps

ln (2 x - 3) = ln (x) - 2 ln (2), x > \frac{3}{2}

Add or subtract both sides

ln (2 x - 3) - (ln (x) - 2 ln (2)) = 0

Subtract the terms

More Steps

ln (\frac{2 x - 3}{x}) + 2 ln (2) = 0

Move the constant to the right-hand side and change its sign

ln (\frac{2 x - 3}{x}) = 0 - 2 ln (2)

Removing 0 doesn't change the value,so remove it from the expression

ln (\frac{2 x - 3}{x}) = - 2 ln (2)

Rewrite the logarithm

ln (\frac{2 x - 3}{x}) = ln (2^{- 2})

Rewrite the expression

\frac{2 x - 3}{x} = 2^{- 2}

Rewrite the expression

\frac{2 x - 3}{x} = \frac{1}{2 ^{2}}

Cross multiply

(2 x - 3) \times 2^{2} = x

Simplify the equation

2^{2} (2 x - 3) = x

Expand the expression

More Steps

8 x - 12 = x

Move the variable to the left side

8 x - 12 - x = 0

Subtract the terms

More Steps

7 x - 12 = 0

Move the constant to the right side

7 x = 0 + 12

Removing 0 doesn't change the value,so remove it from the expression

7 x = 12

Divide both sides

\frac{7 x}{7} = \frac{12}{7}

Divide the numbers

x = \frac{12}{7}

Check if the solution is in the defined range

x = \frac{12}{7}, x > \frac{3}{2}

Solution

x = \frac{12}{7}

Alternative Form

x = 1. \dot{7} 1428 \dot{5}

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