SOLUTION: Solve for x: log5 x + log5(x-24) = 2 Please show me how you got the answer, thanks.

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Question 1130364: Solve for x:
log5 x + log5(x-24) = 2
Please show me how you got the answer, thanks.
Found 3 solutions by josgarithmetic, MathLover1, greenestamps:
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
log%285%2Cx%29%2Blog%285%2C%28x-24%29%29=2

log%285%2C%28x%28x-24%29%29%29=2
log%285%2C%28x%5E2-24x%29%29=2
5%5E2=x%5E2-24x
x%5E2-24x-25=0
%28x-25%29%28x%2B1%29=0
highlight%28x=25%29, but NOT the other solution

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

log%285%2Cx%29+%2B+log%285%2Cx-24%29+=+2............use the rule log%28b%2Cx%2Ay%29=log%28b%2Cx%29%2Blog%28b%2Cy%29

log%285%2Cx%28x-24%29%29+=+2........change the base
log%28x%28x-24%29%29%2Flog%285%29+=+2
log%28x%28x-24%29%29+=+2log%285%29
log%28x%28x-24%29%29+=+log%285%5E2%29...if log same, we have
x%28x-24%29+=+5%5E2
x%5E2-24x+=+25
x%5E2-24x+-25=0........factor
x%5E2%2Bx-25x+-25=0
%28x%5E2%2Bx%29-%2825x+%2B25%29=0
x%28x%2B1%29-25%28x+%2B1%29=0
%28x-25%29%28x+%2B1%29=0

solutions:
if %28x-25%29=0->x=25
if %28x+%2B1%29=0->x=-1-> disregard negative solution; log%28-1%29=i%2Api, then for any negative number
so, your solution is: x=25



Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


log%285%2Cx%29+%2B+log%285%2C%28x-24%29%29+=+2

You have two valid and correct solutions to your equation from other tutors; let me suggest a different method that I personally find easier.

My preferred first step, given two terms with log base 5 on the left and an integer on the right, is to rewrite the integer as a term in log base 5:

log%285%2Cx%29+%2B+log%285%2C%28x-24%29%29+=+log%285%2C25%29

Now, with all terms log base 5, I can convert this to an ordinary algebraic equation using basic rules of logarithms; then solve the equation:

x%28x-24%29+=+25
x%5E2-24x-25+=+0
%28x-25%29%28x%2B1%29+=+0
x+=+25 or x+=+-1

However, the log of a negative number is not a real number, so the only solution is x = 25.