Question 165216
{{{log(5,(x))-log(5,(x-2))=log(5,(4))}}} Start with the given equation



{{{log(5,((x)/(x-2)))=log(5,(4))}}} Combine the logs on the left side using the identity {{{log(b,(A))-log(b,(B))=log(b,(A/B))}}}



{{{(x)/(x-2)=4}}} Since the bases of the logs on both sides are equal, this means that the arguments of the logs (the stuff inside the logs) are equal



{{{x=4(x-2)}}} Multiply both sides by {{{x-2}}}



{{{x=4x-8}}} Distribute.



{{{x-4x=-8}}} Subtract {{{4x}}} from both sides.



{{{-3x=-8}}} Combine like terms on the left side.



{{{x=(-8)/(-3)}}} Divide both sides by {{{-3}}} to isolate {{{x}}}.



{{{x=8/3}}} Reduce.



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Answer:


So the answer is {{{x=8/3}}} which approximates to {{{x=2.667}}}.