Question 129944
{{{log(10,(x-14))-log(10,(5))=log(10,(x-12))-log(10,(x))}}} Start with the given equation



{{{log(10,((x-14)/5))=log(10,((x-12)/x))}}} Combine the logs using the identity {{{log(b,(A))-log(b,(B))=log(b,(A/B))}}}




{{{(x-14)/5=(x-12)/x}}} Raise both sides as exponents with bases of 10 to eliminate the logs on both sides



{{{cross(5)x((x-14)/cross(5))=5cross(x)((x-12)/cross(x))}}} Multiply both sides by {{{5x}}} to eliminate the fractions



{{{x(x-14)=5(x-12)}}} Multiply 



{{{x^2-14x=5x-60}}} Distribute 



{{{x^2-14x-5x+60=0}}}  Subtract 5x from both sides.  Add 60 to both sides. 



{{{x^2-19x+60=0}}} Combine like terms




{{{(x-15)(x-4)=0}}} Factor the left side (note: if you need help with factoring, check out this <a href=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)




Now set each factor equal to zero:

{{{x-15=0}}} or  {{{x-4=0}}} 


{{{x=15}}} or  {{{x=4}}}    Now solve for x in each case



So our possible answers are 

 {{{x=15}}} or  {{{x=4}}} 




However, if you plug in {{{x=4}}} into {{{log(10,(x-14))-log(10,(5))=log(10,(x-12))-log(10,(x))}}} , you'll get a negative number inside the log. So our only solution is {{{x=15}}}




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Answer:


So our answer is {{{x=15}}}