Proofs of 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.




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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.

logarithm properties

The logarithm properties are:

  1. Product Rule
    The logarithm of a product is the sum of the logarithms of the factors.

loga xy = loga x + loga y

  1. Quotient Rule
    The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.

loga = loga x - loga y

  1. Power Rule

loga xn = nloga x

  1. Change of Base Rule



where x and y are positive, 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 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



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