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[Solved] Solve the equation

 
Logorithm
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June 15, 2021 2:41 pm
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Staff 
(@armor)
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Solve the equation below:

2(lnx-1)=ln(x+11).

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June 15, 2021 2:42 pm
Topic starter
Staff 
(@armor)
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We can use the rules for logarithms and rewrite the equation as an exponential term with base \small e. This yields:

\small ln(x-1)^2=ln(x+11)

\small (x-1)^2=x+11

\small x^2-2x+1=x+11

\small x^2-3x-10=0\Rightarrow x_{1}=5,x_{2}=-2.

We have to test whether \small x_{1}  and \small x_{2} are indeed solutions of the given equation. For \small x_{1}=5 we obtain 

\small 2ln(5-1)=ln(5-1)^2=ln16=ln(5+11).

However, for \small x_{2}=-2  the term \small ln(x-1)=ln(-3)  is not defined. Therefore, \small x_{1}=5  is the only solution of the given logarithmic equation.

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