SOLUTION: 6 logx 25 = 7 - log5 x

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Question 1203185: 6 logx 25 = 7 - log5 x
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
+6log%28x%2C25+%29=+7+-+log%285%2Cx%29
+6log%2825+%29%2Flog%28x%29=+7+-+log%28x%29%2Flog%285%29
+6log%2825+%29%2Flog%28x%29=+%287log%285%29+-+log%28x%29%29%2Flog%285%29
+6log%2825+%29log%285%29=+log%28x%29%287log%285%29+-+log%28x%29%29
+6log%285%5E2+%29log%285%29=+log%28x%29%2A7log%285%29+-+log%28x%29%2Alog%28x%29
+6+%2A2log%5E2%285%29=+7log%28x%29%2Alog%285%29+-+log%5E2%28x%29
+log%5E2%28x%29-7log%28x%29%2Alog%285%29%2B12log%5E2%285%29=0
let +log%28x%29=u
+u%5E2-7u%2Alog%285%29%2B12log%5E2%285%29=0

+u=%28-%28-7log%285%29%29%2B-sqrt%2849log%5E2%285%29-48log%5E2%285%29%29%29%2F2
+u=%287log%285%29%2B-sqrt%28log%5E2%285%29%29%29%2F2
+u=%287log%285%29%2B-log%285%29%29%2F2

+u=%287log%285%29%2Blog%285%29%29%2F2=>u=6log%285%29%2F2=>u=3log%285%29
or
+u=%287log%285%29-log%285%29%29%2F2=>u=8log%285%29%2F2=>u=4log%285%29


+log%28x%29=u , then +log%28x%29=3log%285%29 or +log%28x%29=4log%285%29


solutions
+log%28x%29=3log%285%29 ....solve for++x
+log%28x%29=log%285%5E3%29
+x=5%5E3
+highlight%28x=125%29

or
+log%28x%29=4log%285%29.....solve for x
+log%28x%29=log%285%5E4%29
+x=5%5E4
+highlight%28x=625%29



Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


To avoid having to type all those log expressions....

Define y=log%28x%2C5%29

The given equation is

6log%28x%2C25%29=7-log%285%2Cx%29

Note log%28x%2C25%29=log%28x%2C5%5E2%29=2log%28x%2C5%29 and log%285%2Cx%29=1%2Flog%28x%2C5%29

Then the equation is

6%282y%29=7-1%2Fy
12y=7-1%2Fy
12y%5E2=7y-1
12y%5E2-7y%2B1=0
%284y-1%29%283y-1%29=0

The possible solutions are y=1%2F4 and y=1%2F3.

y=1%2F4 --> log%28x%2C5%29=1%2F4 --> x%5E%281%2F4%29=5 --> x=5%5E4=625

y=1%2F3 --> log%28x%2C5%29=1%2F3 --> x%5E%281%2F3%29=5 --> x=5%5E3=125

ANSWERS: x=125 and x=625