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.

**Related Pages**

Common And Natural Logarithm

Rules Of Logarithms

Logarithmic Functions

Rules Of Exponents

Logarithm Rules

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:

- 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

- 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

- Power Rule

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

- 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 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

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

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} = ( 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:**

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

x log _{c} a = log _{c} b

**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

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