Question

Solve. ln(–x + 1) – ln(3x + 5) = ln(–6x + 1) –0.67 or 2 option 1 –1.58 or 0.14 option 2–0.14 or 1.58 option 3–2 or 0.67 option 4

Answer 1

We want to solve

`\ln \mleft(-x+1\mright)-\ln \mleft(3x+5\mright)=\ln \mleft(-6x+1\mright)`

This becomes

`\begin{gathered} \ln (-x+1)-\ln (3x+5)=\ln (-6x+1) \\ \ln (-x+1)-\ln (3x+5)-\ln (-6x+1)=0 \\ \text{From laws of logarithm } \\ \ln ((-x+1))/((3x+5))*(1)/((-6x+1)) \\ \ln ((-x+1))/((3x+5)(-6x+1))=0 \end{gathered}`

Since ln 1 = 0, we have

`\begin{gathered} \ln ((-x+1))/((3x+5)(-6x+1))=\ln 1 \\ \text{Canceling }\ln ,\text{ we have } \\ ((-x+1))/((3x+5)(-6x+1))=1 \\ (-x+1)=(3x+5)(-6x+1) \end{gathered}`

Expanding the equation in the right hand side we have

`\begin{gathered} (-x+1)=(3x+5)(-6x+1) \\ (-x+1)=-18x^2+3x-30x+5 \\ -x+1=-18x^2+3x-30x+5 \\ -18x^2+3x-30x+5=-x+1 \\ -18x^2+3x-30x+x+5-1=0 \\ -18x^2-26x+4=0 \end{gathered}`

Solving the quadratic equation we have

`-18x^2-26x+4=0`

We have

`x=-1.58\text{ or }x=0.14`

Hence, the answer is option 2