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3 ln x  - ln 3 = 3

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mahmodahmad | Student, Grade 9 | eNotes Newbie

Posted October 4, 2010 at 1:57 AM via web

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3 ln x  - ln 3 = 3

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

Posted October 4, 2010 at 1:58 AM (Answer #1)

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3ln x - ln 3 = 3

We know that a ln b = ln b^a

==> ln x^3 - ln 3 = 3

Also, we know that ln a - ln b = ln (a/b)

==> ln (x^3/3) = 3

==> x^3/3 = e^3

==> x^3 = 3e^3

==> x = (3)^1/3 * e

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted October 4, 2010 at 1:58 AM (Answer #2)

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First, we'll impose the constraint of existence of logarithms:

x>0

Now, we'll use the power property of logarithms, so that:

3 ln x = ln x^3

We'll re-write the equation:

 ln x^3  - ln 3 = 3

Since the bases of logarithms are matching, we'll use the quotient rule:

ln (x^3/3) = 3

We'll have:

x^3/3 = e^3

x^3 = 3e^3

x = e*[(3)^1/3]

Since the solution is positive, it is valid.

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neela | High School Teacher | (Level 3) Valedictorian

Posted October 4, 2010 at 2:04 AM (Answer #3)

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3lnx -ln3 = 3

Solution;

3lnx - ln3 =lne^3 by definition

3lnx = ln3+lne^3.

3lnx = ln3*e^3. as lna+lnb = lnab

lnx^3 = ln3*e^3, as m ln a = ln a^m

Take antilog of both sides:

x^3 = 3e^3

x = (3e^3)^(1/3)

x = {(3^(1/3)}e

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phucnguyen08 | Student, College Freshman | (Level 1) Honors

Posted October 4, 2010 at 2:38 AM (Answer #4)

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lnx^3-ln3=3        ln(x^3/3)=3

x^3/3=e^3     x^3=3e^3           x=e*3^1/3

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