Logarithm Properties



In this lesson, 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 properrties.

The rules of logarithms are

1) Product Rule

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

loga xy = loga x + loga y

2) Quotient Rule

The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator

loga = loga x – loga y

3) Power Rule

loga xn = nloga x

4) Change of Base Rule

where x and y are postive, and a > 0, a ≠ 1


Proof for the Product Rule

loga xy = loga x + loga y

Proof:

Step 1:
Let m = loga x and n = loga y

Step 2: Write in exponent form
x = am and y = an

Step 3: Multiply x and y
x
y = am an = am+n

Step 4: Take log a of both sides and evaluate
log a xy = log a am+n
log a xy = (m + n) log a a
log a xy = m + n
log a xy = loga x + loga y


Proof for the Quotient Rule

loga = loga x – loga y

Proof:

Step 1:
Let m = loga x and n = loga y

Step 2: Write in exponent form
x = am and y = an

Step 3: Divide x by y
x
÷ y = am ÷ an = am – n

Step 4: Take log a of both sides and evaluate
log a (x ÷ y) = log a am – n
log a (x ÷ y) = (m – n) log a a
log a (x ÷ y) = m – n
log a (x ÷ y) = loga x – loga y


Proof for the Power Rule

loga xn = nloga x

Proof:

Step 1:
Let m = loga x

Step 2: Write in exponent form
x = am

Step 3: Raise both sides to the power of n
xn
= ( am )n

Step 4: Convert back to a logarithmic equation
log a xn = mn

Step 5: Substitute for m = loga x
log a xn = n loga x


Proof for the Change of Base Rule

Proof:

Step 1:
Let x = loga b

Step 2: Write in exponent form
ax = b

Step 3: Take log c of both sides and evaluate
log c ax = log c b
x
log 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 BA and
Quotient Rule: log A - log B = log (A/B)





Proof of the logarithm property
Change of Base Rule: loga B = log x B/ log x A





OML Search

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

Find Tutors

OML Search