Proofs of Logarithm Properties

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.

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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
x log _c a = log _c b

 

Videos: Proof of the logarithm properties
Proof of Product Rule: log A + log B = log AB

• Show Video Lesson

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

• Show Video Lesson

Proof of Change of Base Rule: log_a B = log_x B/ log_x A

• Show Video Lesson

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