Definition of Logarithms
You have surely already studied powers and know that, for example:
But suppose you want to find a power to which to raise the number 10 so that the result is 10,000,000. That can be written as follows:
Could you solve for the letter '
' in that equation?
The equation we wrote is an exponential equation. To be able to solve for the variable '' we need to use a logarithm. A logarithm is an "operation" or "function" that returns the power to which you must raise a given base to obtain a desired result. In our example, the base is 10 and the desired result is 10000000, so we can write:
In general, we can express logarithmic notation as follows:
Where:
a is the base
x is the desired result (also known as the argument)
y is the power to which base a is raised
Below, we show you some examples of expressions in exponential notation and logarithmic notation:
It should be noted that the most commonly used bases in logarithms are 
y 
(Euler's number, 
)
When we use base it is not necessary to write the base of the logarithm:
The logarithm with base is known as the Napierian logarithm (or natural logarithm) and is represented as:
Properties of Logarithms
1 The logarithm of a product equals the sum of the logarithms of the factors
2 The logarithm of a quotient equals the difference of the logarithm of the dividend and the logarithm of the divisor
3 The logarithm of a power equals the product of the exponent times the logarithm of the base
4 The logarithm of a root equals the quotient between the logarithm of the radicand and the index of the root
From properties 
and 
we can deduce that:
5 The logarithm base '
' of '' is '
'
6 The logarithm of is 
(regardless of the logarithm base)
Thus:
7 The argument of a logarithm must always be greater than zero
For 
it holds that 
Use of Logarithm Properties
Base Changes
To write a logarithm of base '
' in an equivalent expression with logarithm of base '' we can do the following:
Let
We can rewrite the expression in its exponential notation as:
Applying 
on both sides of the equality:
Applying property and solving for '' we obtain:
Therefore:
Example: Rewrite 
in 
Applying:
Solve an Expression with Combined Operations Applying the Properties of Logarithms
Example: Solve the operation 
applying the properties of logarithms.
Let's equate the expression we want to solve to '':
Since all numbers are powers of 
, we can apply on both sides:
Applying the properties of logarithms on the right side we obtain:
Solving the logarithms:
Rewriting in exponential notation:
Thus:
Write an Expression Containing Operations with Logarithms as an Expression Containing a Single Logarithm
Example: Write the following operation with logarithms as an expression with a single logarithm
Applying the properties of logarithms:
Summarize with AI:
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