All real number solutions of equation log_(2011)(2010x) = log_(2010)(2011x) are in certain interval. Which one is it?

ajanlr

ajanlr

Answered question

2022-10-29

All real number solutions of equation log 2011 ( 2010 x ) = log 2010 ( 2011 x ) are in certain interval. Which one is it?
This task has to be done with no calculator, but I don't have basic idea how to start. Can someone give me advice, I know this is pretty easy but I need direction for particularly this one? EDIT: I do it like this
log 2011 ( 2010 x ) = log 2010 ( 2011 x )
ln 2010 x ln 2011 = ln 2011 x ln 2010
ln 2010 + ln x ln 2011 = ln 2011 + ln x ln 2010
ln 2 2010 + ln x ( ln 2010 ) = ln 2 2011 + ln x ( ln 2011 )
ln 2 2010 ln 2 2011 ln x ( ln 2011 ln 2010 ) = 0
( ln 2011 ln 2010 ) ( ln 2011 + ln 2010 ln x ) = 0
ln ( 2010 2011 ) x = ln 1
x = 2010 2011
What am I doing wrong here? And also, how to even get solution (which is x ( 0 , 1 2011 ] )

Answer & Explanation

Cagliusov8

Cagliusov8

Beginner2022-10-30Added 15 answers

Hint:
log 2011 ( 2010 x ) = ln 2010 + ln x ln 2011
Rewrite log 2010 ( 2011 x ) similarly and solve for ln x
Marley Meyers

Marley Meyers

Beginner2022-10-31Added 3 answers

x = 1 2010 × 2011 Just use normal properties of log like
1) first change the base of every log to e
2) expand l n 2010 x and l n 2011 x using property l n a b = l n a + l n b
3) solve for x

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