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log 2 x - log 4 x = 2    find x

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tuleentoto | Student, Grade 11 | (Level 1) eNoter

Posted October 4, 2010 at 1:45 AM via web

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log 2 x - log 4 x = 2    find x

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neela | High School Teacher | (Level 3) Valedictorian

Posted October 4, 2010 at 1:54 AM (Answer #1)

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log2x-log4x = 2.

Therefore

log(2) x - log(4) x = 2

We convert log(4)x  to log2(x)/log(2)4  to ve the same base.

log(2)x - log(2)x/log(2)4 = 2

log(2)x -(1/2)log(2)x = 2

(1-1/2) logx = 2

(1/2) log(2)x = 2

log(2)x = 4

 x = 2^4 = 16

 

 

 

 

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted October 4, 2010 at 1:46 AM (Answer #2)

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If the base of logarithms is 10 ,so the  bases are matching, we'll use the quotient property:

log 2 x - log 4 x = 2

log 2x/4x = 2

We'll simplify:

log 1/2 = 2

1/2 = 10^2 impossible for any value of x!

But if the bases are 2 and 4, we'll solve the equation:

log 2 x = log 4 x*log 4 2

log 2 x = log 4 x / log 2 4

log 2 x = log 4 x/2

log 4 x = 2 log 2 x

We'll use the power property of logarithms:

log 4 x = log 2 x^2

We'll substitute log 4 x by the  log 2 x^2.

log 2 x - log 2 x^2 = 2

We'll use the quotient property:

log 2 x/x^2 = 2

log 2 1/x = 2

1/x = 2^2

1/x = 4

x = 1/4

Since the solution is positive, it is accepted.

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hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted October 4, 2010 at 1:46 AM (Answer #3)

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log 2 x - log 4 x = 2

Let us rewirte:

We know that:

log a b = log c b/ log c a

==> log 4 x = log 2 x/ log 2 4

                    = log 2 x / 2

                   = (1/2)*log 2 x

==> log 2 x - (1/2) log 2 (x) = 2

==> (1/2) log 2 (x) = 2

No multiply by 2:

==> log 2 (x) = 4

==> x= 2^4

==> x= 16

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