Question

Solve for x. ln(2x-2) - ln(x-1)= ln x

Answer 1

Remember the following property of logarithms:

$\ln (a)-\ln (b)=\ln (\frac{a}{b})$

Use this property to simplify the left member of the equation and solve for x:

$\begin{gathered} \ln (2x-2)-\ln (x-1)=\ln (x) \\ \Rightarrow\ln (\frac{2x-2}{x-1})=\ln (x) \\ \Rightarrow\frac{2x-2}{x-1}=x \\ \Rightarrow2x-2=x(x-1) \\ \Rightarrow2(x-1)=x(x-1) \end{gathered}$

Since the factor (x-1) appears at both sides of the equation, we can simplify it as long as x is different from 1 (since the factor should not be equal to 0). Then:

$x=2$

Plug in x=2 into the original equation to check the solution:

$\begin{gathered} \ln (2(2)-2)-\ln (2-1)=\ln (2) \\ \Rightarrow\ln (4-2)-\ln (1)=\ln (2) \\ \Rightarrow\ln (2)-\ln (1)=\ln (2) \end{gathered}$

Since ln(1)=0, then:

$\ln (2)=\ln (2)$

Therefore, the solution is x=2.