Question

What is a Logarithm? (Tutorial)

Answer 1

What is a logarithm?

A logarithm is, simply put, how to find how many times you have to multiply one number by itself to get another number. Here’s an example.

How many times do you have to multiply 3 by itself to get 81?
\[3*3*3*3=81\]
Therefore,
\[\log_{3}(81)=4 \]
Let’s dissect this expression a bit.
\[\log_{3}\]
The little 3 after the log, called the base, is the number you have to multiply by itself. So,
\[\log_{2}\]
is how many times you have to multiply 2 by itself to get another number. We put the other number that we are trying to solve for in parentheses like this:
\[\log_{2}(64)\]
So, this just means, “How many times we have to multiply 2 by itself to get 64?”, and the answer to that is 6, because
\[2*2*2*2*2*2=64\]
So,
\[\log_{2}(64)=6\]

Answer 2

Another way we can look at this is in exponential terms. One way to rephrase the expression
\[\log_{9}(59,049)\]
is “What power do we have to raise 9 to to get 59,049?”
\[9^5=59,049\]
So,
\[\log_{9}(59,094)=5\]
A more general way to quickly understand logarithms is to see the crossovers, illustrated below:

Logarithms can also be decimals:
\[\log_{10}(50)=1.69897\]
(approximately) because
\[10^1.69897=50\]
What about if we work backwards? How do we find the expression below?
\[\log_{10}(0.001)\]
Well, the only way to move backwards is to divide!
\[1/10/10/10=0.001\]
But, we didn’t multiply! Isn’t that breaking one of the rules of logarithms? Actually,
\[10^-3=0.001\]
So,
\[\log_{10}(0.001)=-3\]

Answer 3

There are also other ways to write logarithms, for example, common logarithms are written without a base.
\[\log(1000)\]
When you encounter a common logarithm, never fear, this just means the base is 10. These are used quite a bit because a base of 10 is actually very common.
\[\log(1000) = \log_{10}(1000) = 3\]
Another way to write logarithms are natural logarithms, which always have a base of e. (e = Euler’s number = 2.71828…) These are written using ln instead of log.
\[\ln(54.598...) = \log_{e}(54.598...) = 4\]

Answer 4

nice work, great tutorial!

First to answer, first to medal, first to say nice work, great tutorial! :)

Answer 5

sure

Answer 6

Thanks so much!

Answer 7

Cool! ur awesome at explaining it!

Answer 8

Once again, thank you!

Answer 9

This is an amazing tutorial! Great Job :)

Answer 10

Thanks :D

Answer 11

Absolutely no problem! Its really amazing and explained very well!

Answer 12

good work!

Answer 13

heya great work! o^_^o
some tips you can use fraction like http://prntscr.com/91jfio
and for exponent \[10^{1.69897}\] base^{exponent}

you can add the formula how to convert log to an exponential form i guess that might help students to understand how u got \[10^{1.69897}=50\]

Answer 14

good job! keep it up!

Answer 15

Thank you all :D

Answer 16

exponents.

good one

Answer 17

I think it's a good tutorial, but there is a huge lack of theorical background.
It's not like a logarithm is an intuitive idea, it's a definition that was studied under some years before reaching what we currently know as "logarithm".

Answer 18

Hi all. I'm now realizing part of the tutorial is missing.

Answer 19

Posting it now.

Answer 20

Another way we can look at this is in exponential terms. One way to rephrase the expression
\[\log_{9}(59,049)\]
is “What power do we have to raise 9 to to get 59,049?”
\[9^5=59,049\]
So,
\[\log_{9}(59,094)=5\]
A more general way to quickly understand logarithms is to see the crossovers, illustrated below:

Answer 21

There we go.

Answer 22

I always just think log => exponent , nice little write up review

Answer 23

Yup.

Answer 24

@Owlcoffee This is more of an introductory tutorial... although I completely agree with you. Logarithms are just easier ways to write exponents.

Answer 25

Mathematicians are just lazy. :P

Answer 26

Logarithm is Exponent

It may not be exactly the initial development of the concept, but that is where it landed.