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## Algebra (all content)

### Course: Algebra (all content)>Unit 11

Lesson 22: Introduction to logarithms (Algebra 2 level)

# Intro to logarithms

Sal explains what logarithms are and gives a few examples of finding logarithms. Created by Sal Khan.

## Want to join the conversation?

• is there any synonym of logarithm?
• Logarithm is based on the combination of two Greek words: logos and arithmos (number). Logos (λόγος) is a rather curious Greek word with multiple meanings. In this case, you could translate it as "ratio" or "proportion". The word "logarithm" was invented by John Napier in 1614.
• Who invented logarithms? And for what reason?
• Sir John Napier did... It was actually for ease of calculations, when we didn't have digital calculators. Logarithms made it easy for people to carry out otherwise difficult operations, eg: find the value of 4th root of 24. we can simply take log(24) and divide by 4. The antilog of the resultant figure will give us the answer. This is quite a feat, considering that we are not using any calculator!
• In what grade do you learn logarithims?
• Technically, since logarithms are part of Algebra 2, by regular USA standards, probably 11th grade. Like some people have said, you can obviously learn it whenever you want, since it isn't a very complicated subject, but if you are on the average math pacing in the USA, you should learn in somewhere in 11th grade. Some people take Algebra 2 earlier than that though, so you may learn it in grades 8, 9 or 10 if you find yourself skilled and interested in math.
• Is log base 1 of 1 equal to 1, 0, or both?
• Log base 1 of 1 would have infinite answers not just 1 and 0
• what is zero to the zero power?
• That depends on whom you ask:
There are several very important mathematical theorems that require 0⁰ = 1. Thus, many or most mathematicians have simply defined 0⁰ = 1 without formal proof.

There are a few mathematicians who disagree and assert that 0⁰ is indeterminate or undefined. Particularly in certain constructions that come up in calculus (but not at this level of study) 0⁰ is a form that is indeterminate (for which there are ways to work around it and find a solution).

Thus, it really just depends on whom you ask. Most likely, your instructor would assert 0⁰ = 1, but you should ask her to see, to make sure she is not one of those who dissent from the majority of mathematicians.

NOTE: To date, no mathematician has ever been able to formally prove what, if anything, 0⁰ equals. The assertion that it equals 1 is done so that exceptions do not have to be made to those important theorems.
• Why x+(1/x) >=2 if x is positive and x+(1/x)<=-2 if x is negative real number?
(1 vote)
• First, let's consider the case that x is positive. Note that sqrt(x) is real and nonzero since x is positive.
To show x + (1/x) >=2, it is enough to show that x + (1/x) - 2 >= 0.
The trick is to express x + (1/x) - 2 as a perfect square trinomial!
x + (1/x) - 2 = [sqrt(x)]^2 + [1/sqrt(x)]^2 - 2
= [sqrt(x)]^2 + [1/sqrt(x)]^2 - 2sqrt(x)*[1/sqrt(x)]
= [sqrt(x)]^2 - 2sqrt(x)*[1/sqrt(x)] + [1/sqrt(x)]^2
= [sqrt(x) - 1/sqrt(x)]^2
>=0 since sqrt(x) - 1/sqrt(x) is real (because x is positive).
So x + (1/x) - 2 >= 0, which then implies x + (1/x) >= 2.

Now, let's consider the case that x is negative. Then -x is positive. From the result of the first case using -x in place of x, we have
-x + [1/(-x)] >= 2 which implies -x - (1/x) >= 2 which implies -[x + (1/x)] >= 2 which implies
x + (1/x) <= -2.

Have a blessed, wonderful day!
• Can a logarithm be written without a base?
• Writing a log without a base is implied that it has base 10.
log(100) = log_10(100) = 2

The other "exception" is the natural log, which you don't write with a base.
ln(x) is the same form as log_e(x), but ln is more perferred.
• lets say they ask you a question that has log 100,000 and it equals 5 how do you get the answer without using the calculator
• There is a rule: log10^n = n
So write your problem as log 10^5, this equals 5.