Logarithm Properties

Logarithm Properties

Basic Properties of Logarithms

Like exponents, logarithms also have several important properties that need to be understood. These properties will be very helpful in solving various logarithmic problems.

Let $a > 0$ and $a \neq 1$ , $b > 0$ , $c > 0$ , $m > 0$ , $m \neq 1$ , where $a, b, c, m, n$ are real numbers $(a, b, c, m, n \in \mathbb{R})$ . The following are logarithm properties:

$^a\log a = 1$
$^a\log 1 = 0$
$^a\log a^n = n$

Proving Logarithm Properties

Logarithm of Multiplication

Property $4$ : $^a\log (b \times c) = ^a\log b + ^a\log c$

Proof: Let $^a\log b = m$ and $^a\log c = n$

This means:

$b = a^m$ and $c = a^n$

Using the property of exponents:

b \times c = a^m \times a^n = a^{m+n}

Therefore:

^a\log (b \times c) = ^a\log (a^{m+n}) = m + n = ^a\log b + ^a\log c

Logarithm of Division

Property $5$ : $^a\log \left(\frac{b}{c}\right) = ^a\log b - ^a\log c$

Proof: Let $^a\log b = m$ and $^a\log c = n$

Then $b = a^m$ and $c = a^n$

Recall that $\frac{a^m}{a^n} = a^{m-n}$ , so:

\frac{b}{c} = \frac{a^m}{a^n} = a^{m-n}

Therefore:

^a\log \left(\frac{b}{c}\right) = ^a\log (a^{m-n}) = m - n = ^a\log b - ^a\log c

Logarithm of Power

Property $6$ : $^a\log b^n = n \cdot ^a\log b$

Proof: Let $^a\log b = m$

$^a\log b^n$ means the logarithm of $b$ raised to the power of $n$

^a\log b^n = ^a\log (\underbrace{b \times b \times b \times \ldots \times b}_{n \text{ factors}})

Using property $4$ repeatedly:

^a\log b^n = \underbrace{^a\log b + ^a\log b + ^a\log b + \ldots + ^a\log b}_{n \text{ factors}} = n \cdot ^a\log b

Change of Base

Property $7$ : $^a\log b = \frac{^m\log b}{^m\log a} = \frac{1}{^b\log a}$

Proof: Based on the definition of logarithm, $^a\log b = c$ if and only if $b = a^c$

Suppose we use base $m$ for the logarithm of $b$ :

^m\log b = ^m\log a^c

Using property $6$ :

^m\log b = c \cdot ^m\log a

Since $c = ^a\log b$ , then:

^m\log b = ^a\log b \cdot ^m\log a

Therefore:

^a\log b = \frac{^m\log b}{^m\log a}

If $m = b$ , then:

^a\log b = \frac{^b\log b}{^b\log a} = \frac{1}{^b\log a}

Chain Rule for Logarithms

Property $8$ : $^a\log b \times ^b\log c = ^a\log c$

Proof: Based on the definition:

^a\log b = m \Leftrightarrow b = a^m

Substitute the value of $b$ into the equation for $c$ :

c = b^n = (a^m)^n = a^{mn}

Since $b$ , then:

^a\log c = ^a\log (a^{mn}) = mn = ^a\log b \times ^b\log c

Application Example

Suppose we want to calculate $^5\log 125$ .

Using property $6$ :

^5\log 125 = ^5\log 5^3 = 3 \cdot ^5\log 5 = 3 \cdot 1 = 3

Exercises

Simplify the following expressions:
1. $^9\log 81$
2. $^2\log 64 - ^2\log 16$

Answer Key

Determining logarithm values
1. Answer:
  $^9\log 81 = ^9\log 9^2$