Logarithms are a very convenient mathematical tool, used extensively to simplify calculations. In our experience, many students know how to use logarithms (logs for short), but do not know what they actually are. Therefore, this part should be read keeping in mind the aim that you need to understand the underlying idea behind logs before trying to put them to use.
Consider the following equation:
This would be read as “Ten to the power two equals one hundred”. With logs, you can equivalently state this problem as:
which would be read as “The logarithm of hundred to the base ten is two.”
Consider some more examples:
Read as: “The log of to base is ” 
Read as: “The log of to base is ” 
Read as: “The log of to base is ” 
Read as: “The log of to base is ” 
Thus, we see that the log of any number to a given base is the power to which the base must be raised in order to equal the given number. In other words,
if 
or in words,
if 
Thus, logs are just a way of expressing relations involving powers.
As an exercise, show that
We’ll now discuss some properties that logs satisfy:
(1)
This follows by definition.
(2) is defined only for and . We’ll try to understand why cannot equal :
raised to any power will always equal
for all  
If we try to calculate we see that there can be no value possible for this log, since no matter what power you raise to, you’ll never obtain . Thus, is not a valid base for logs.

(3) for any base . This is because any number raised to power is , so that
(4) Log of a product
One of the most useful properties of logs is this:
that is,. the log of a product is equal to the sum of the individual logs. Let us first see the justification of this property.
Assume
Thus, notice carefully that this property is nothing but a manifestation of the fact that when two exponential terms with the same base are multiplied, their powers add, as shown in . This is something you must always keep in mind.
Why is this property so useful? Well, it lets us reduce large multiplications to additions. More on this soon.
(5) Log of a dfraction
Analogous to the product property, we’ve the division property:
As the alert reader might be expecting, this property is a result of the fact that when two exponential terms with the same base are divided, their powers subtract. Let us prove this property explicitly now.
Assuming and we have
As with the multiplication property, this property lets us convert divisions into subtractions and simplifies calculations, as we’ll soon see.
(6) Log of an exponential term
This property states that
that is, the log of to any base is times the log of to the same base. This is again straightforward to justify:
Assume
Thus, this property is a consequence of the fact that when an exponential term is raised to another power, the two powers multiply.
We’ll now understand how the last three properties help us with calculations.
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