Logarithms

Definition

Note that this only works because we have restricted the values of and . The base and argument must be positive real numbers.

Property

The logarithm is bijective on

Graph

left=-1; right=11;
bottom=-3; top=3;
---
y = \log_{10}(x)

Natural Log

Definition

(see Euler's Number)

Log Laws

Constant Laws

Law

For all :

Proof

By definition of log with . Therefore
By definition of log with . Therefore

Limit Laws

Law

For all

(see limit)

Proof

Let :

Power Law

Law

For all :

Proof

Let .

Let .
Then by definition of log, .
Taking the exponent to both sides gives .
Using the definition of log again, , and substituting , we have

Product and Quotient Laws

Law

For all :

Proof

Let

Let and
Then by definition of log and , and we have
Taking log of both sides:

\log_{a}(bc) & = \log_{a}(a^{n + m}) \
& = (n + m) \log_{a}(a) && \text{by power law} \
& = n + m && \text{by constant law} \
& = \log_{a}(b) + \log_{a}(c)
\end

\begin{align}
\log_{a}\left( \frac{b}{c} \right) & = \log_{a}(a^{n + m}) \
& = (n + m) \log_{a}(a) && \text{by power law} \
& = n + m && \text{by constant law} \
& = \log_{a}(b) - \log_{a}(c)
\end

Other Laws

Log as Exponent with Same Base

Law

For all :

More generally, for all :

Proof

Let .

Then by power rule.
By definition of log, we can write .

Log as Exponent with Different Base

Law

For all :

Proof

Let .
Then such that .
Putting an exponent on both sides gives .
Then by "log as exponent with same base" law, we have .
Substituting , we have .

Log Base with Exponent

Law

For all :

Proof

Let .
Then such that .
We can write this as which gives .
Substituting , we have .

Change of Base

Law

For all :

where can be any base

Proof

Let :
Let . Then .

For all :