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:
loga xy = loga x + loga y
loga = loga x - loga y
loga xn = nloga x
where x and y are positive, and a > 0, a ≠ 1
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
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
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:
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 properties
Proof of Product Rule: log A + log B = log AB
Proof of Power Rule: Alog B = log BA and
Proof of Quotient Rule: log A - log B = log (A/B)
Proof of Change of Base Rule: loga B = logx B/ logx A
Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.