3.6 Properties of Logarithms

Before we learn about logarithmic functions, we will outline the properties of logarithms like we did for exponents. There are nine basic properties of logarithms. Let $u$ and $v$ be real numbers with $u>0$ and $v>0$. Moreover, let $a$ and $b$ be real numbers with $a>0$, $a\ne 1$, $b>0$, and $b\ne 1$.

1. Definition of a Logarithm

The expression

$v={\log }_{a}u$

is read as “$v$ is the logarithm base $a$ of $u$,” if and only if $u=a^v$. The word logarithm is often shortened to log. Note that $u>0$ , so u has to be a positive number.

Here are a few examples:

${\log }_{5}25=2⟺25=5^2$

${\log }_3\frac{1}{3}=-1⟺\frac{1}{3}=3^{-1}$

${\log }_3\frac{1}{81}=-4⟺\frac{1}{9}=3^{-4}$

${\log }_2\sqrt{2}=\frac{1}{2}⟺\sqrt{2}=2^{\frac{1}{2}}$

${\log }_{4}4=1⟺4=4^1$

${\log }_{7}1=0⟺1=7^0$

2. Common and Natural Logarithms

Logarithms to base 10 are called common logarithms. For simplicity, ${\text{log}}_{10}u$ is abbreviated as $\text{log}u$, without writing the base. The most commonly used type of logarithms are natural logarithms, that is, logarithms to base $e$. The conventional notation of natural logarithms is $\ln u$ instead of ${\log }_{e}u$.

3. Trivial Identities of Logarithms

There are two trivial identities of logarithms. The first is that a logarithm to the base $a$ of 1 is 0. Mathematically,

${\log }_{a}1=0$

The second is that a logarithm to the base $a$ of $a$ is 1. Mathematically,

${\log }_{a}a=1$

4. Power Rule of Logarithms

A logarithm to the base $a$ of some exponential number $u^r$ is the product of $r$ and the logarithm to the base $a$ of $u$.

${\log }_a(u^r)=r\cdot {\log }_{a}u$

Note that ${\log }_a(u^r)\ne ({\log }_{a}u{)}^r$ when $u\ne 1$ and $u\ne a$.

Here are a few examples.

${\log }_2(5^6)=6{\log }_{2}5$

${\log }_5({13}^{-3})=-3{\log }_{5}13$

${\log }_3\sqrt{5}={\log }_3{5}^{\frac{1}{2}}=\frac{1}{2}{\mathrm{ log}}_{3}5$

5. Canceling Logarithms

Given Property 4 above, a logarithm to the base $a$ of some exponent power of $a$, $a^r$, is $r$.

${\log }_a(a^u)=u$

This is because ${\log }_{a}a=1$:

${\log }_a(a^u)=u\cdot {\log }_{a}a$
$=u\cdot 1$
$=u$

Moreover,

$a^{{\log }_{a}u}=u$

This is because, based on the definition of a logarithm,

$a^{{\log }_{a}u}=u⟺{\log }_{a}u=lo{g}_{a}u$

6. Logarithm of a Product

The logarithm of a product is the sum of the logarithms of the factors.

${\log }_a(u\cdot v)={\log }_{a}u+{\log }_{a}v$

7. Logarithm of a Quotient

The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.

${\log }_a\left(\frac{u}{v}\right)={\log }_{a}u-{\log }_{a}v$

8. Change of Base

The base of a logarithm can be changed from $a$ to $b$ with the following formula:

${\log }_{a}u=\frac{{\log }_{b}u}{{\log }_{b}a}$

9. Taking Logarithms on Both Sides

Suppose that

$u=v$

Then the equation holds that if both sides of equation take a logarithm of the same base,

${\log }_{b}u={\log }_{b}v$