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arrow_back Solve the logarithmic equation \(\ln (x-2)+\ln (2 x-3)=2 \ln x\)

by Bronze Status
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Solve the logarithmic equation \(\ln (x-2)+\ln (2 x-3)=2 \ln x\)

1 Answer

Best answer
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Best answer
(1) Use Power Rule: \(\log _{b} x^{c}=c \log _{b} x\).
\[
\ln (x-2)+\ln (2 x-3)=\ln \left(x^{2}\right)
\]
(2) Use Product Rule: \(\ln (x y)=\ln x+\ln y\).
\[
\ln ((x-2)(2 x-3))=\ln \left(x^{2}\right)
\]
3 Cancel "In" on both sides.
\[
(x-2)(2 x-3)=x^{2}
\]
(4) Expand.
\[
2 x^{2}-3 x-4 x+6=x^{2}
\]

5) Simplify \(2 x^{2}-3 x-4 x+6\) to \(2 x^{2}-7 x+6\).
\[
2 x^{2}-7 x+6=x^{2}
\]
(6) Move all terms to one side.
\[
2 x^{2}-7 x+6-x^{2}=0
\]
(7) Simplify \(2 x^{2}-7 x+6-x^{2}\) to \(x^{2}-7 x+6\).
\[
x^{2}-7 x+6=0
\]
8) Factor \(x^{2}-7 x+6\).
\[
(x-6)(x-1)=0
\]

9 Solve for \(x\).
\[
x=6,1
\]
10 Check solution
When \(x=1\), the original equation \(\ln (x-2)+\ln (2 x-3)=2 \ln x\) does not hold true. We will drop \(x=1\) from the solution set.
11 Therefore,
\[
x=6
\]
by Bronze Status
(7,476 points)

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