Jump To Content
Algebra

Community

Algebra

Primary Navigation

Logarithms- basics

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.

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 on logs may be dangerous by pfly truck by pingnews.com Vancouver by ahisgett

magdaaltman
  • Authority 27
Post Body
magdaaltman said:

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….

  • Quote
  • Posted 4 months ago.
magdaaltman
  • Authority 27
Post Body
magdaaltman said:

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….

  • Quote
  • Posted 4 months ago.
  • Your comment will be modifiable for 10 minutes after posted.

Page Author

Avatar
oLahav
Name
oLahav

From Here You Can…

Information

Most Recent Related Content

Published In…

Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 Unported License.