SOLUTION: Hi. I'm am required to find x in this equation: {{{log (5, (5-4x))}}} = {{{log (sqrt(5), (2-x))}}}. My teacher told me answer is +1 or -1. I do know the formula - Change of Base La

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Hi. I'm am required to find x in this equation: {{{log (5, (5-4x))}}} = {{{log (sqrt(5), (2-x))}}}. My teacher told me answer is +1 or -1. I do know the formula - Change of Base La      Log On


   

Question 135430: Hi. I'm am required to find x in this equation: log+%285%2C+%285-4x%29%29 = log+%28sqrt%285%29%2C+%282-x%29%29. My teacher told me answer is +1 or -1. I do know the formula - Change of Base Law - to solve the equation, but the problem is, I do not know how to apply the formula due to the square root of 5 as the base of the equation.
I've tried by using the formula to change log+%285%2C+%285-4x%29%29:
log+%28sqrt%285%29%2C+%285-4x%29%29/log+%28sqrt%285%29%2C+sqrt%285%29%29 = log+%28sqrt%285%29%2C+%282-x%29%29
log+%28sqrt%285%29%2C+%285-4x%29%29/1 = log+%28sqrt%285%29%2C+%282-x%29%29
5-4x = 2-x
-3x = -3
x = 1
However, I don't know how to find -1. Please teach me the correct method. Thanks.
(By the way, the answer is +1 with the + sign begin underlined. I hope you understand what I mean. And is this answer the same as 1 and -1? Thanks.)
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Solve for x:
log%285%2C%285-4x%29%29+=+log%28sqrt%285%29%2C%282-x%29%29
Change the base of the second logarithm to that of the first using:log%28b%2C%28M%29%29+=+%28log%28a%2C%28M%29%29%29%2F%28log%28a%2C%28b%29%29%29
=%28log%285%2C%282-x%29%29%29%2F%281%2F2%29log%285%2C%285%29%29=2%2Alog%285%2C%282-x%29%29
So, we end up with:
log%285%2C%285-4x%29%29+=+2%2Alog%285%2C%282-x%29%29 Applying the power rule to the right side:
log%285%2C%285-4x%29%29+=+log%285%2C%282-x%29%5E2%29 therefore:
5-4x+=+%282-x%29%5E2 Simplifying:
5-4x+=+4-4x%2Bx%5E2 Add 4x to both sides.
5+=+4%2Bx%5E2 Subtract 5 from both sides.
0+=+x%5E2-1 Factor the right side.
%28x%2B1%29%28x-1%29+=+0 so...
x+=+1 or x+=+-1
The answer can be read as: x = + or - 1