SOLUTION: Solve for x: {{{log(4,(x+13))+log(4,(x+1))=3}}}

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Question 130690: Solve for x:


log%284%2C%28x%2B13%29%29%2Blog%284%2C%28x%2B1%29%29=3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
log%284%2C%28x%2B13%29%29%2Blog%284%2C%28x%2B1%29%29=3 Start with the given equation.


log%284%2C%28x%2B13%29%28x%2B1%29%29=3 Combine the logs using the identity log%28b%2C%28A%29%29%2Blog%28b%2C%28B%29%29=log%28b%2C%28A%2AB%29%29


4%5E3=%28x%2B13%29%28x%2B1%29 Rewrite the equation using the property: log%28b%2C%28x%29%29=y ====> b%5Ey=x

64=%28x%2B13%29%28x%2B1%29 Evaluate 4%5E3 to get 64

64=x%5E2%2B14x%2B13 Foil the right side


0=x%5E2%2B14x%2B13-64 Subtract 64 from both sides.

0=x%5E2%2B14x-51 Combine like terms


%28x%2B17%29%28x-3%29=0 Factor the left side (note: if you need help with factoring, check out this solver)



Now set each factor equal to zero:
x%2B17=0 or x-3=0

x=-17 or x=3 Now solve for x in each case


So our possible solutions are

x=-17 or x=3


However, if we plug in x=-17, we'll have negative numbers in the argument of the logs. Since you cannot take the log of a negative number, the value x=-17 is an extraneous solution (ie it does not satisfy the original equation).



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Answer:

So our only solution is x=3