Logarithm Properties

In this lesson, we will look at the four properties of logarithms and their proofs. You may also want to look at the lesson on how to use the logarithm properrties.

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 xⁿ = nlog_a x

4) Change of Base Rule

where x and y are postive, 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 = aⁿ

Step 3: Multiply x and y
x • y = a^m • aⁿ = 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 = aⁿ

Step 3: Divide x by y
x ÷ y = a^m ÷ aⁿ = 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 xⁿ = 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
xⁿ = ( a^m )ⁿ

Step 4: Take log _a of both sides and evaluate
log _a xⁿ = log _a a^mn
log _a xⁿ = mn log _a a
log _a xⁿ = mn
log _a xⁿ = 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

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