To derive the Properties of Logarithms, we can start by defining two auxiliary general logarithmic expressions.

a = l o g_{b} m c = lo g_{b} n

Now, from the definition of logarithm we know that there is an equivalent exponential representation for a logarithmic expression.

y = b^{x} \Leftrightarrow x = lo g_{b} y

We can use this to rewrite our auxiliary logarithmic expressions in exponential form.

a = l o g_{b} m c = lo g_{b} n ⇕ b^{a} = m b^{c} = n

This way, we can alter them using the Properties of Exponents. Once we have simplified by using the Properties of Exponents, we can rewrite our result using the corresponding logarithmic expressions. We will now work out some examples.

Deriving the Product Property of Logarithms

Using the expressions introduced before, lets find the product of

m

and

n .

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

▼

Simplify and rewrite using logarithms

MultPow

$a^{m} \cdot a^{n} = a^{m + n}$

m n = b^{a + c}

Rewrite

Rewrite $m n = b^{a + c}$ as $lo g_{b} (m n) = a + c$

lo g_{b} (m n) = a + c

SubstituteII

$a = l o g_{b} m$ , $c = l o g_{b} n$

lo g_{b} (m n) = l o g_{b} m + l o g_{b} n

This last result is known as the Product Property of Logarithms.

$lo g_{b} (m n) = lo g_{b} m + lo g_{b} n$

Deriving the Quotient Property of Logarithms

Using the expressions introduced above, we will now find the quotient of

m

and

n .

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

▼

Simplify and rewrite using logarithms

DivPow

$\frac{a ^{m}}{a ^{n}} = a^{m - n}$

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

Rewrite

Rewrite $\frac{m}{n} = b^{a - c}$ as $lo g_{b} (\frac{m}{n}) = a - c$

lo g_{b} (\frac{m}{n}) = a - c

SubstituteII

$a = l o g_{b} m$ , $c = l o g_{b} n$

lo g_{b} (\frac{m}{n}) = l o g_{b} m - l o g_{b} n

This last result is known as the Quotient Property of Logarithms.

$lo g_{b} (\frac{m}{n}) = lo g_{b} m - lo g_{b} n$

Deriving the Power Property of Logarithms

Let's consider the logarithm base

b

of

m^{n} .

Since we know that

m = b^{a},

we can use it in our expression.

lo g_{b} (m^{n}) = lo g_{b} ((b^{a})^{n})

▼

Simplify and rewrite using logarithms

PowPow

$(a^{m})^{n} = a^{m \cdot n}$

lo g_{b} (m^{n}) = lo g_{b} (b^{a n})

$lo g_{b} (b^{m}) = m$

lo g_{b} (m^{n}) = a n

Substitute

$a = l o g_{b} m$

lo g_{b} (m^{n}) = (l o g_{b} m) n

Multiply

lo g_{b} (m^{n}) = n lo g_{b} m

This last result is known as the Power Property of Logarithms.

$lo g_{b} (m^{n}) = n lo g_{b} m$

Exercise