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Concept
List of logarithmic identities
Summary
In mathematics, many logarithmic identities exist. The following is a compilation of the notable of these, many of which are used for computational purposes.
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By definition, we know that:
where or .
Setting ,
we can see that:
So, substituting these values into the formula, we see that:
which gets us the first property.
Setting ,
we can see that:
So, substituting these values into the formula, we see that:
which gets us the second property.
Many mathematical identities are called trivial, only because they are relatively simple (typically from the perspective of an experienced mathematician). This is not to say that calling an identity or formula trivial means that it's not important.
Logarithms and exponentials with the same base cancel each other. This is true because logarithms and exponentials are inverse operations—much like the same way multiplication and division are inverse operations, and addition and subtraction are inverse operations.
Both of the above are derived from the following two equations that define a logarithm:
(note that in this explanation, the variables of and may not be referring to the same number)
Looking at the equation , and substituting the value for of
we get the following equation:
which gets us the first equation.
Another more rough way to think about it is that ,
and that that "" is .
Looking at the equation
and substituting the value for of , we get the following equation:
which gets us the second equation.
Another more rough way to think about it is that ,
and that that something "" is .
Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below. The first three operations below assume that x = bc and/or y = bd, so that logb(x) = c and logb(y) = d. Derivations also use the log definitions x = blogb(x) and x = logb(bx).
Official source
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