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.

The following table gives a summary of the logarithm properties. Scroll down the page for more explanations and examples on how to proof the logarithm properties.

The logarithm properties are 1) Product Rule

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

**Proof for the Product Rule**

**Proof for the Quotient Rule**

**Proof for the Power Rule**

**Proof for the Change of Base Rule**

**Videos: Proof of the logarithm properties**

Proof of Product Rule: log A + log B = log AB Proof of Power Rule: Alog B = log B^{A} and

Proof of Quotient Rule: log A - log B = log (A/B) Proof of Change of Base Rule: log_{a} B = log _{x} B/ log
_{x} A

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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

The following table gives a summary of the logarithm properties. Scroll down the page for more explanations and examples on how to proof the logarithm properties.

The logarithm properties 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= log_{a}- log_{a}x_{a}y

3) Power Rule

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

4) Change of Base Rule

where* x *and *y* are positive, and *a*
> 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 Product Rule: log A + log B = log AB Proof of Power Rule: Alog B = log B

Proof of Quotient Rule: log A - log B = log (A/B) Proof of Change of Base Rule: log

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You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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