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Natural Log Problem

  1. Oct 14, 2009 #1
    1. The problem statement, all variables and given/known data
    y and x are both close to 1.
    [tex]ln(y) - ln(x) \approx (y-x)(lnx)'[/tex]

    Can someone explain me why this is true?

    By the way, (lnx)' is the derivative of lnx, which is just 1/x

    3. The attempt at a solution

    I guess this is something to do with the taylor series of ln(x). I tried to expand ln(y) and ln(x) by taylor series, but I did not get anything from there.

    [tex]((y-1)-\frac{(y-1)^{2}}{2}+\frac{(y-1)^{3}}{3}...)-((x-1)-\frac{(x-1)^{2}}{2}+\frac{(x-1)^{3}}{3}...)[/tex]
     
    Last edited: Oct 14, 2009
  2. jcsd
  3. Oct 14, 2009 #2

    Mark44

    Staff: Mentor

    I would say it's closer to the limit definition of ln(x), using a difference quotient.

    [tex]d/dx(ln(x))~=~\lim_{h \rightarrow 0} \frac{ln(x + h) - ln(x)}{h}[/tex]
    For suitably small h,
    [tex]d/dx(ln(x))~\approx~ \frac{ln(x + h) - ln(x)}{h}[/tex]

    Now, let y = x + h, and you're almost where you want to go.

    BTW, this should be in the Calculus and Beyond section, not Precalculus.
     
  4. Oct 15, 2009 #3
    No it's algebra

    ln(x^3) - ln(x) = 10
    then x=e^[something]

    I don't know how to solve for it
    and would also like to know the algebraic approach to solving
    ln(x^n) - ln(x) = some integer
    then x = e^[some thing]
     
  5. Oct 15, 2009 #4

    Mark44

    Staff: Mentor

    If this is an algebra problem, how do you explain the derivative in it?
     
  6. Oct 15, 2009 #5
    I have no idea you tell me,
    All I know is, that it was in my algebra test - university first year mathematics

    and i'm pretty sure my university wouldn't accidently put a calculus question in an algebra exam

    you're probably right though, there is a calculus and an algebraic approach
    and seeing as this was in a first year algebra exam the awnser would be simple and elegant to get too
     
  7. Oct 15, 2009 #6

    Mentallic

    User Avatar
    Homework Helper

    I took another look at the problem after getting some inspiration from this line of yours :smile:

    ok, simple and elegant? Yes, it struck me.
    [tex]ln(y) - ln(x) \approx (y-x)(lnx)'[/tex]

    [tex]RHS=\frac{y-x}{x}[/tex] since [tex](lnx)'=\frac{1}{x}[/tex] (yes, derivatives aren't precalculus, but this one was simple)

    [tex]=\frac{y}{x}-1[/tex]

    therefore, [tex]ln(y)-ln(x) \approx \frac{y}{x}-1[/tex]

    [tex]ln\left(\frac{y}{x}\right) \approx \frac{y}{x}-1[/tex]

    [tex]\frac{y}{x} \approx e^{\frac{y}{x}-1}[/tex]

    For [itex]x,y\approx 1[/itex]

    [tex]\frac{y}{x}\approx 1[/tex]

    [tex]e^{\frac{y}{x}-1}\approx e^{1-1} \approx 1[/tex]

    Is this sufficient?
     
  8. Oct 15, 2009 #7

    Mark44

    Staff: Mentor

    From your first post, what you were supposed to do was show that ln(y) - ln(x) can be approximated by (y - x) times the derivative of the ln function. Once you have shown that, you're done.
    e1 - 1 = e0 = 1 exactly. These are not approximations.
    It was unnecessary. What was sufficient was showing that ln(y) - ln(x) [itex]\approx[/itex] (y - x)(ln(x))'. After you have done what a problem asks, it's fine to continue exploring different avenues for your own enlightenment or enjoyment, but the problem doesn't require this.
     
    Last edited: Oct 15, 2009
  9. Oct 15, 2009 #8

    Mark44

    Staff: Mentor

    Then maybe you are mistaken about what is being presented in your university class. I don't know where your university is, but in many U.S. universities, the first-year, college-level (not remedial) math classes start with calculus or precalculus, with the assumption being that a student took algebra and geometry and trig in high school. The straight algebra classes are essentially remedial classes, with class numbers below the 100 level. That's how it is in Washington state, where I live, and I believe that this is true also in many other states.
    There is no straight algebraic approach to this problem, because the concept of the derivative is not part of algebra - it is strictly a calculus concept.
     
  10. Oct 15, 2009 #9
    I apologize for posting this on a wrong section. But thank you guys, I perfectly understand it now.
     
  11. Oct 15, 2009 #10

    Mentallic

    User Avatar
    Homework Helper

    Yes e0=1 exactly, but the fact that the question says for x,y[itex]\approx[/itex]1 then y/x[itex]\approx[/itex]1 therefore ey/x-1[itex]\approx[/itex]e1-1

    no?
     
  12. Oct 16, 2009 #11

    Mark44

    Staff: Mentor

    Sure, that part is an approximation.
     
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