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.

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_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 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 = 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: Convert back to a logarithmic equation
log _a xⁿ = mn

Step 5: Substitute for m = log_a x
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 properties
Proof of Product Rule: log A + log B = log AB

• Show Step-by-step Solutions

Proof of Power Rule: Alog B = log B^A and
Proof of Quotient Rule: log A - log B = log (A/B)

• Show Step-by-step Solutions

Proof of Change of Base Rule: log_a B = log _x B/ log _x A

• Show Step-by-step Solutions

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