Reference

Properties of Logarithms

Rule

Product Property of Logarithms

The logarithm of a product can be written as the sum of the individual logarithms of each factor.

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

This property is only valid for positive values of

b,

m,

and

n,

and for

b \neq = 1 .

As an example, the expression

lo g_{3} (7 \cdot 4)

can be rewritten using this property.

lo g_{3} (7 \cdot 4) = lo g_{3} 7 + lo g_{3} 4

Proof

Start by recalling the definition of a logarithm.

lo g_{b} a = c \Leftrightarrow a = b^{c}

Before proving the desired property, two other identities will be justified. The first equation of the definition states that

c = lo g_{b} a .

Therefore

lo g_{b} a

can be substituted for

c

in the second equation.

a = b^{c} Substitute a = b^{l o g_{b} a}

Furthermore, the second equation states that

a

is equal to

b^{c} .

This means that

b^{c}

can be substituted for

a

in the first equation.

lo g_{b} a = c Substitute lo g_{b} b^{c} = c

The obtained identities will be used together with the Product of Powers Property to prove the Product Property of Logarithms.

lo g_{b} m n

Rewrite

Rewrite $m n$ as $m \cdot n$

lo g_{b} (m \cdot n)

$m = b^{l o g_{b} (m)}$

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

MultPow

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

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

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

lo g_{b} m + lo g_{b} n

Rule

Quotient Property of Logarithms

The logarithm of a quotient can be written as the difference between the logarithm of the numerator and the logarithm of the denominator.

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

This property is valid for positive values of

b,

m,

and

n,

and for

b \neq = 1 .

For example, the expression

lo g_{3} \frac{7}{4}

can be rewritten using this property.

lo g_{3} \frac{7}{4} = lo g_{3} 7 - lo g_{3} 4

Proof

Start by recalling the definition of a logarithm.

lo g_{b} a = c \Leftrightarrow a = b^{c}

Before proving the desired property, two other identities will be justified. The first equation of the definition states that

c = lo g_{b} a .

Therefore

lo g_{b} a

can be substituted for

c

in the second equation.

a = b^{c} Substitute a = b^{l o g_{b} a}

Furthermore, the second equation states that

a

is equal to

b^{c} .

This means that

b^{c}

can be substituted for

a

in the first equation.

lo g_{b} a = c Substitute lo g_{b} b^{c} = c

The obtained identities will be used together with the Quotient of Powers Property to prove the Quotient Property of Logarithms.

lo g_{b} \frac{m}{n}

$m = b^{l o g_{b} (m)}$

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

DivPow

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

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

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

lo g_{b} m - lo g_{b} n

Rule

Power Property of Logarithms

The logarithm of a power can be written as the product of the exponent and the logarithm of the base.

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

This property is valid for positive values of

b,

m,

and

n,

and for

b \neq = 1 .

For example,

lo g_{2} 7^{4}

can be rewritten using this property.

lo g_{2} 7^{4} = 4 lo g_{2} 7

Proof

Start by recalling the definition of a logarithm.

lo g_{b} a = c \Leftrightarrow a = b^{c}

Before proving the desired property, two other identities will be justified. The first equation of the definition states that

c = lo g_{b} a .

Therefore

lo g_{b} a

can be substituted for

c

in the second equation.

a = b^{c} Substitute a = b^{l o g_{b} a}

Furthermore, the second equation states that

a

is equal to

b^{c} .

This means that

b^{c}

can be substituted for

a

in the first equation.

lo g_{b} a = c Substitute lo g_{b} b^{c} = c

The obtained identities will be used together with the Power of a Power Property to prove the Power Property of Logarithms.

lo g_{b} m^{n}

$m = b^{l o g_{b} (m)}$

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

PowPow

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

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

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

(lo g_{b} m) \cdot n

CommutativePropMult

Commutative Property of Multiplication

n lo g_{b} m

Rule

Change of Base Formula

A logarithm of arbitrary base can be rewritten as the quotient of two logarithms with the same base by using the change of base formula.

$lo g_{c} a = \frac{lo g _{b} a}{lo g _{b} c}$

This rule is valid for positive values of $a, b,$ and $c,$ where $b$ and $c$ are different than $1 .$

Proof

Let

x = lo g_{c} a .

Therefore, by the definition of a logarithm, it is known that

a = c^{x} .

x = lo g_{c} a \Leftrightarrow a = c^{x}

By the Reflexive Property of Equality,

lo g_{b} a

is equal to itself.

lo g_{b} a = lo g_{b} a

Substitute

$a = c^{x}$

lo g_{b} a = lo g_{b} c^{x}

$lo g_{b} (a^{m}) = m \cdot lo g_{b} (a)$

lo g_{b} a = x lo g_{b} c

Substitute

$x = l o g_{c} a$

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

DivEqn

$LHS / lo g_{b} c = RHS / lo g_{b} c$

\frac{lo g _{b} a}{lo g _{b} c} = lo g_{c} a

RearrangeEqn

Rearrange equation

lo g_{c} a = \frac{lo g _{b} a}{lo g _{b} c} ✓

Rule

Inverse Properties of Logarithms

A logarithm and a power with the same base undo each other.

$lo g_{b} b^{x} = x and b^{l o g_{b} x} = x$

In particular, the above equations also hold true for common and natural logarithms.

lo g 1 0^{x} = x ln e^{x} = x and and 1 0^{l o g x} = x e^{l n x} = x

Proof

The general equations will be proved one at a time.

$lo g_{b} b^{x} = x$

This identity can be proved by using the Power Property of Logarithms and the definition of a logarithm.

lo g_{b} b^{x}

$lo g_{b} (a^{m}) = m \cdot lo g_{b} (a)$

x lo g_{b} b

$lo g_{b} (b) = 1$

x (1)

IdPropMult

Identity Property of Multiplication

x ✓

The logarithm of

b^{x}

with base

b

is equal to

x .

$b^{l o g_{b} x} = x$

Let

lo g_{b} x = a .

Therefore, by the definition of a logarithm,

b^{a} = x .

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

This will be used to prove the identity.

b^{l o g_{b} x}

Substitute

$lo g_{b} x = a$

b^{a}

Substitute

$b^{a} = x$

x ✓

Therefore,

b

to the power of

lo g_{b} x

is equal to

x .

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Proof

Proof

Proof

Proof

Proof

logb​bx=x

blogb​x=x

$lo g_{b} b^{x} = x$

$b^{l o g_{b} x} = x$