List of logarithmic identities
compilation of logarithm related identities
In mathematics, many logarithmic identities exist. The following is a compilation of the notable of these, many of which are used for computational purposes.
Trivial identities
Trivial mathematical identities are relatively simple (for an experienced mathematician), though not necessarily unimportant. The trivial logarithmic identities are as follows:
Explanations
By definition, we know that:
log
b
(
y
)
=
x
⟺
b
x
=
y
,
{\displaystyle \log _{b}(y)=x\iff b^{x}=y,}
where
b
≠
0
{\displaystyle b\neq 0}
and
b
≠
1
{\displaystyle b\neq 1}
.
Setting
x
=
0
{\displaystyle x=0}
,
we can see that:
b
x
=
y
⟺
b
(
0
)
=
y
⟺
1
=
y
⟺
y
=
1
{\displaystyle b^{x}=y\iff b^{(0)}=y\iff 1=y\iff y=1}
So, substituting these values into the formula, we see that:
log
b
(
y
)
=
x
⟺
log
b
(
1
)
=
0
,
{\displaystyle \log _{b}(y)=x\iff \log _{b}(1)=0,}
which gets us the first property.
Setting
x
=
1
{\displaystyle x=1}
, we can see that:
b
x
=
y
⟺
b
(
1
)
=
y
⟺
b
=
y
⟺
y
=
b
{\displaystyle b^{x}=y\iff b^{(1)}=y\iff b=y\iff y=b}
So, substituting these values into the formula, we see that:
log
b
(
y
)
=
x
⟺
log
b
(
b
)
=
1
,
{\displaystyle \log _{b}(y)=x\iff \log _{b}(b)=1,}
which gets us the second property.
Cancelling exponentials
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:
b
log
b
(
x
)
=
x
because
antilog
b
(
log
b
(
x
)
)
=
x
{\displaystyle b^{\log _{b}(x)}=x{\text{ because }}{\mbox{antilog}}_{b}(\log _{b}(x))=x}
log
b
(
b
x
)
=
x
because
log
b
(
antilog
b
(
x
)
)
=
x
{\displaystyle \log _{b}(b^{x})=x{\text{ because }}\log _{b}({\mbox{antilog}}_{b}(x))=x}
Both of the above are derived from the following two equations that define a logarithm: (note that in this explanation, the variables of
x
{\displaystyle x}
and
x
{\displaystyle x}
may not be referring to the same number)
log
b
(
y
)
=
x
⟺
b
x
=
y
{\displaystyle \log _{b}(y)=x\iff b^{x}=y}
Looking at the equation
b
x
=
y
{\displaystyle b^{x}=y}
, and substituting the value for
x
{\displaystyle x}
of
log
b
(
y
)
=
x
{\displaystyle \log _{b}(y)=x}
, we get the following equation:
b
x
=
y
⟺
b
log
b
(
y
)
=
y
⟺
b
log
b
(
y
)
=
y
,
{\displaystyle b^{x}=y\iff b^{\log _{b}(y)}=y\iff b^{\log _{b}(y)}=y,}
which gets us the first equation.
Another more rough way to think about it is that
b
something
=
y
{\displaystyle b^{\text{something}}=y}
,
and that that "
something
{\displaystyle {\text{something}}}
" is
log
b
(
y
)
{\displaystyle \log _{b}(y)}
.
Looking at the equation
log
b
(
y
)
=
x
{\displaystyle \log _{b}(y)=x}
, and substituting the value for
y
{\displaystyle y}
of
b
x
=
y
{\displaystyle b^{x}=y}
, we get the following equation:
log
b
(
y
)
=
x
⟺
log
b
(
b
x
)
=
x
⟺
log
b
(
b
x
)
=
x
,
{\displaystyle \log _{b}(y)=x\iff \log _{b}(b^{x})=x\iff \log _{b}(b^{x})=x,}
which gets us the second equation.
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