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College Algebra

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I REALLY don't understand the reason/basis/use of logarithms. I have listened to my teacher, who never is clear on much of anything (it would help if he spoke better English), and a more advanced student, who couldn't explain them to me.
The problem I am working on is: log8(1000x), where 8 is the base. We are supposed to pull apart the multiplication of the 1000x and then solve the parts we are able to solve without a calculator.

So far, I have: log8(1000) + log8(x). the answer is supposed to be 3+logx
how does log8(1000) =3, and how does log8(x) change to just logx?

Shouldn't log8(1000) be the same as 8^y=1000? If so, then 8^3 does not equal 1000. So why does the book say it is 3?

Also, what is the purpose of e and ln in trig?

  • College Algebra -

    The purpose of logs is that when multiplying, logs add. Thence the slide rule, and a mydrid of mechanical computing devices. The other use, is that many natural processes in nature are logarithimic in nature (population growth, interest, radioactiveity, in fact, anything whose increase depends on the amount already present is logarithimic). Number of telephone calls overseas (or Maine) is dependent on the number of telephones there.

    log8(1000)+log8(x)

    well, log8(1000) is only 3 if the digits 1000 is base 8, which I doubt.
    1000base8 = no ones+no eights+no 64s + one 512 (8^3).

    1000base8 = 512base10
    This is a screwball example to teach logs...and bothers me that a book would use it. In my text writing, I would have said log8(1000base8)=3, but frankly, I wouldn't have ever mixed logs with other base systems.

  • College Algebra -

    The purpose of e and ln.

    It turns out, in natural processes, which are continual in nature, have a form of expontential (ie log) growth to a base e (2.71828....). So it is a handy base. Humans leared to count to number bases which were whole numbers (based on 10, or 8, or 4, or 16, and in the case of Mayans, 5, or 20. Some Babolynians used 12 as a base.

    But e is God's choice, mainly.

    Radioactive decay is e^kt, population is base e, in fact, any continous process is base e.

    Ln is the log to base e. So log 100 to base 10 is 2, but ln 100=4.605
    So get use to it, and put the blame on God. Frankly, in the physical sciences, e and ln are bases to use.

  • College Algebra -

    ok - the e and ln make sense to me. On the second part of the equation - log8x - how come the base 8 is just dropped and listed as logx? Doesn't that change the equation to just any base? So any logz(x) can just be changed to logx?

  • College Algebra -

    No, it can't, it is sloppy math.

    you cant say log8(1000)=log1000=3
    log8(x) is not logx unless you know the log is base 8.

    for instance, if x is 64
    log8(x)=2
    log (64)=1.806

  • College Algebra -

    a car travels 280 miles at a certain speed. If the speed had been 5mph faster, the trip would have taken 1 hr less. find the speed.

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