2) Quotient Rule
The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator
loga3) Power Rule
loga xn = nloga x4) Change of Base Rule
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
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
xlog c a =
log c b
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