# Logarithm Properties

In these lessons, we will look at the four properties of logarithms
and their proofs. They are the product rule, quotient rule, power
rule and change of base rule.

You may also want to look at the lesson on how to use the logarithm properties.

Related Topics:

More Algebra Lessons

Free Math Worksheets

The rules of logarithms are

1) Product Rule

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

log_{a} *xy* = log_{a} *x* +
log_{a} y

2) Quotient Rule

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

log_{a} = log_{a}
x - log_{a} *y*

3) Power Rule

log_{a} *x*^{n} = *n*log_{a}
x

4) Change of Base Rule

where* x *and *y* are positive, and *a*
> 0, *a* ≠ 1

## Proof for the Product Rule

log_{a} *xy* = log_{a} *x* +
log_{a} y

**Proof:**

**Step 1: **

Let *m* = log_{a} *x* and *n = *log_{a}
y

**Step 2:**
Write in exponent form

*x* = *a*^{m }and *y* = *a*^{n}

**Step 3: **Multiply*
x * and *y*

x *y* = *a*^{m } *a*^{n}
= *a*^{m+n}

**Step 4:** Take
log _{a} of both sides and evaluate

log _{a} xy = log _{a} *a*^{m+n
}log _{a} xy = (*m + n*) log
_{a} *a*

log _{a} xy = *m + n*

log _{a} xy = log_{a} *x* +
log_{a} y

## Proof for the Quotient Rule

log_{a} = log_{a}
x - log_{a} *y*

**Proof:**

**Step 1: **

Let *m* = log_{a} *x* and *n = *log_{a}
y

**Step 2:**
Write in exponent form

*x* = *a*^{m }and *y* = *a*^{n}

**Step 3: **Divide*
x * by *y*

x ÷ *y* = *a*^{m }÷ *a*^{n}
= *a*^{m - n}

**Step 4:** Take
log_{ a} of both sides and evaluate

log _{a} (*x* ÷ *y*)* = *log
_{a} *a*^{m - n
}log _{a} (*x* ÷ *y*)*
= *(*m - n*) log _{a} *a*

log _{a} (*x* ÷ *y*)* =
**m - n*

log _{a} (*x* ÷ *y*)* =*
log_{a} *x* - log_{a} y

## Proof for the Power Rule

log_{a} *x*^{n} = *n*log_{a}
x

**Proof:**

**Step 1: **

Let *m* = log_{a} *x*

**Step 2:**
Write in exponent form

*x* = *a*^{m }

**Step 3: **Raise
both sides to the power of *n*

x^{n} = ( *a*^{m })^{n}

**Step 4:**
Convert back to a logarithmic equation

log _{a} *x*^{n}* = mn*

**Step 5:**
Substitute for *m* = log_{a} *x*

log _{a} x^{n} = n log_{a}
*x*

## Proof for the Change of Base Rule

**Proof:**

**Step 1: **

Let *x* = log_{a} *b*

**Step 2:**
Write in exponent form

*a*^{x } = b

**Step 3: **Take
log_{ c} of both sides and evaluate

log _{c} *a*^{x} = log _{c}
b^{
}xlog* *_{c} *a* =
log* *_{c} b

## Videos

Proof of the logarithm property:

Product Rule: log A + log B = log AB

Proofs of the logarithm properties:

Power Rule: Alog B = log B^{A} and

Quotient Rule: log A - log B = log (A/B)

Proof of the logarithm property

Change of Base Rule: log_{a} B = log _{x} B/ log
_{x} A