Question

Logarithm Tutorial
Creator of Tutorial: malcolmmcswain

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

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

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: