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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

User David Lemphers
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1 Answer

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19 votes

SOLUTION

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

User Anudeep
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