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In this lesson, 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:

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The rules of logarithms are
**Proof for the Product Rule **

**Proof for the Quotient Rule **

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}y2) Quotient Rule

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

log

= log_{a}- log_{a}x_{a}y3) Power Rule

log

_{a}x=^{n}nlog_{a}x4) Change of Base Rule

where

xandyare positive, anda> 0,a≠ 1

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

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

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

log * _{a} xy = *log

log

= log_{a}- log_{a}x_{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

**Step 3: **Divide*
x * by *y**
x* ÷

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

log * _{a} *(

log

_{a}x=^{n}nlog_{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}* = (

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

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

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

**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

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

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