Mathematics

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Logarithms- basics

by Oren Lahav

The Logarithms



Introducing- logs

Logarithms, which we'll refer to as logs, are an important mathematical concept. Logs are very simply a way to represent a mathematical relation. They're a lot loss scary than you may think. No reason to freak out.

Let's go over basic stuff- how can I represent the number 4? Well, there are infinitely many ways to do so. 1+3 is one way. 2 * 2 is another. So is \frac{12}{3}, and 2 ^2. And so is \log_{2} 16.

Oh no! Freak out! Ahh!

No, that's a joke. See, logs are just one more way to say 4, or 20, or other numbers. In that case, you may think logs are useless. Well, let's look at the formal definition:

If x = b ^y, then y=\log_b x. So for example, we all know that 16=2 ^4. Thus, 4=\log_2 16. See, it really is simple! So \log_5 125=3 and \log_4 4096=6. %

%{font-family:verdana;font-size:13px}Logs are super useful when you want to know something like this- for what x will 2 ^x = 600? That's simple, for x=\log_2 600, which you can find out using a scientific calculator. The answer will be something like 9.22. So you see, logs are really useful in some situations.

Log rules! (literally)

There are a few important log rules you should know:

1. \log_a a ^x = x, which makes sense because a ^x=a ^x.

2. \log_a (x * y)=\log_a x+\log_a y. This is because a ^x * a ^y = a ^{(x+y)}.

3. \log_a \frac{x}{y} = \log_a x - \log_a y.

4. \log_a x ^n= n\log_a x (most useful rule!)

5. \log_a x=\frac{\log_b x}{\log_b a} (change of base formula).

The change of base formula is extra-useful because calculators usually give only \log_{10} results (generally speaking, log with no base specified is taken as log base 10).

Also, note the domain of the inside of the log. If we have \log_a x, then x > 0. This makes intuitive sense looking at the basic definition of the log.

Let's do some practice!

Ex: Solve for x, if \log_7 (x ^3 + 27) - \log_7 (x+3) = 2.

Super easy! Let's first use rule 3. We have: \log_7 \frac{x ^3 + 27}{x+3}=2. Notice that by cube laws, x ^3+27=(x+3)(x ^2 -3x+9), so we're left with \log_7 (x ^2-3x+9)=2.

Now, apply the general log principle. What this really means is that 7 ^2=x ^2-3x+9, so we continue solving x ^2-3x-40=0 and find that x=8 or x=-5.

Finally, we note that since \log_7 -5+3 is undefined, x can't be -5. So our answer is x=8!

Logs are fun!

That's pretty much all you need to know. You can practice some simple log problems on the log test. Good luck!

That's it for logs!

Photo Credits:

Back on the Log Train, by Claire L. Evans, Standing on logs may be dangerous by pfly, Logging truck by pingnews.com, Logs Vancouver by ahisgett

13 Comments
    mathforallgrades
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    mathforallgradesSat, 03 Mar 2012 12:41:53 -0000

    For an excellent discussion on logarithms, please click on:

    www.math-for-all-grades.com

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    akshayhallur
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    Akshay Hallur RFri, 27 Jan 2012 04:35:28 -0000

    Excellent work..

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    aruntechgeek
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    aruntechgeekTue, 06 Sep 2011 15:43:06 -0000

    very good tutorial,

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    Raj Gauswami
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    Raj GauswamiSat, 23 Jul 2011 11:29:28 -0000

    Thanks a lot ,,,,
    I was so confused about these all but now i think i can go ahead with this slogans and all ,,

    :))

    But Maths Sucks :((

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    bright_star
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    bright_starTue, 12 Apr 2011 05:46:00 -0000

    Helpful .. THANK YOU

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    satyak
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    satyakTue, 08 Mar 2011 10:26:04 -0000

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    HEYITZME
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    HEYITZMETue, 05 Oct 2010 15:33:58 -0000

    Thnx ::::)))))))))

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    dewanganhk
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    dewanganhkSun, 13 Jun 2010 15:19:08 -0000

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    luvy
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    LuvyWed, 28 Apr 2010 05:08:14 -0000

    it's a job well done.

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    lalchand87
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    LalchandSat, 13 Mar 2010 18:42:32 -0000

    Thanks a lot

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    Upal Roy
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    Upal RoyMon, 21 Sep 2009 08:32:34 -0000

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    magdaaltman
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    Mon, 25 Aug 2008 18:59:23 -0000

    nicely done! although after taking the well constructed test, I see I may have to review a little (still at the 'standing on logs may be dangerous' phase….

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    magdaaltman
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    Mon, 25 Aug 2008 18:59:22 -0000

    nicely done! although after taking the well constructed test, I see I may have to review a little (still at the 'standing on logs may be dangerous' phase….

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oLahav
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About: I don't know how to describe myself... besides, I'm way too biased in this particular topic. What's the point?

Last Updated At Dec 07, 2012
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