Properties of Logarithms
Properties of Logarithms
Properties of Logarithms Since logarithms are the inverse of exponents, their properties are similar.
Exponential Function
x b
=y
Logarithmic Function
x = logb y
Logarithm of a Product Rule 1: The logarithm of a product is equal to the summation of the logarithms of its factors.
Recall:
Product Rule of Exponents
bx Therefore:
×
by =
b
x+y
Logarithm of a Product
logb (x × y) = logb x + logb y
Logarithm of a Product Rule 1: The logarithm of a product is equal to the summation of the logarithms of its factors.
log3x
=log3 + logx
log3 + log2 =log (3 × 2) =log6
Logarithm of a Product Rule 1: The logarithm of a product is equal to the summation of the logarithms of its factors.
log3x
=log3 + logx
log3 + log2 =log (3 × 2) =log6
Logarithm of a Quotient Rule 2: The logarithm of a quotient is equal to the difference of the logarithms of the dividend and divisor.
Recall:
Quotient Rule of Exponents
bx x–y = b y b Therefore:
Logarithm of a Quotient
x logb = logb x – logb y y
Logarithm of a Quotient Rule 2: The logarithm of a quotient is equal to the difference of the logarithms of the dividend and divisor.
log
x 4
=logx – log4
log9 – log3 =log =log3
9 3
Logarithm of a Quotient Rule 2: The logarithm of a quotient is equal to the difference of the logarithms of the dividend and divisor.
log
x 4
=logx – log4
log9 – log3 =log =log3
9 3
Fully expand the expression:
log
5x 2y
Simplify the expression following expression: What is this as a single logarithm?
log8 + log2 – log4 a. b. c. d.
log8 log6 log4 log2
What is this expression as a single logarithm?
log8 + log2 – log4
Logarithm of a Power Rule 3: The logarithm of a power is equal to the logarithm of the base of the power times the exponent.
Recall:
Power of a Power Rule
(bx)y Therefore:
= b
x×y
Logarithm of a Power
logb (x
y)
= ylogb x
Logarithm of a Power Rule 3: The logarithm of a power is equal to the logarithm of the base of the power times the exponent.
log2x =xlog2
3log5 3 =log5
=log125
Logarithm of a Power Rule 3: The logarithm of a power is equal to the logarithm of the base of the power times the exponent.
log2x =xlog2
3log5 3 =log5
=log125
Simplify thevalue following What is the of x?expression:
8 = logx4 a. b. c. d.
2 100 254 4096
What is the value of x?
8 = logx4
Express as a single logarithm with a coefficient of 1:
𝟒 𝒍𝒐𝒈 𝒙 – 𝟑 𝒍𝒐𝒈 𝒚 + 𝟓 𝒍𝒐𝒈 𝒛
Fully expand the expression:
𝒍𝒐𝒈𝟒(𝟑𝒙 𝟓 𝒚)
Simplify theexact following What is the valueexpression: of the expression?
3log24 a. b. c. d.
3 6 18 32
What is the exact value of the expression?
3log24
Properties of Logarithms Since logarithms are the inverse of exponents, their properties are similar.
Exponential Function
x b
=y
Logarithmic Function
x = logb y
Logarithm of a Product Rule 1: The logarithm of a product is equal to the summation of the logarithms of its factors.
Recall:
Product Rule of Exponents
bx Therefore:
×
by =
b
x+y
Logarithm of a Product
logb (x × y) = logb x + logb y
Logarithm of a Product Rule 1: The logarithm of a product is equal to the summation of the logarithms of its factors.
log3x
=log3 + logx
log3 + log2 =log (3 × 2) =log6
Logarithm of a Product Rule 1: The logarithm of a product is equal to the summation of the logarithms of its factors.
log3x
=log3 + logx
log3 + log2 =log (3 × 2) =log6
Logarithm of a Quotient Rule 2: The logarithm of a quotient is equal to the difference of the logarithms of the dividend and divisor.
Recall:
Quotient Rule of Exponents
bx x–y = b y b Therefore:
Logarithm of a Quotient
x logb = logb x – logb y y
Logarithm of a Quotient Rule 2: The logarithm of a quotient is equal to the difference of the logarithms of the dividend and divisor.
log
x 4
=logx – log4
log9 – log3 =log =log3
9 3
Logarithm of a Quotient Rule 2: The logarithm of a quotient is equal to the difference of the logarithms of the dividend and divisor.
log
x 4
=logx – log4
log9 – log3 =log =log3
9 3
Fully expand the expression:
log
5x 2y
Simplify the expression following expression: What is this as a single logarithm?
log8 + log2 – log4 a. b. c. d.
log8 log6 log4 log2
What is this expression as a single logarithm?
log8 + log2 – log4
Logarithm of a Power Rule 3: The logarithm of a power is equal to the logarithm of the base of the power times the exponent.
Recall:
Power of a Power Rule
(bx)y Therefore:
= b
x×y
Logarithm of a Power
logb (x
y)
= ylogb x
Logarithm of a Power Rule 3: The logarithm of a power is equal to the logarithm of the base of the power times the exponent.
log2x =xlog2
3log5 3 =log5
=log125
Logarithm of a Power Rule 3: The logarithm of a power is equal to the logarithm of the base of the power times the exponent.
log2x =xlog2
3log5 3 =log5
=log125
Simplify thevalue following What is the of x?expression:
8 = logx4 a. b. c. d.
2 100 254 4096
What is the value of x?
8 = logx4
Express as a single logarithm with a coefficient of 1:
𝟒 𝒍𝒐𝒈 𝒙 – 𝟑 𝒍𝒐𝒈 𝒚 + 𝟓 𝒍𝒐𝒈 𝒛
Fully expand the expression:
𝒍𝒐𝒈𝟒(𝟑𝒙 𝟓 𝒚)
Simplify theexact following What is the valueexpression: of the expression?
3log24 a. b. c. d.
3 6 18 32
What is the exact value of the expression?
3log24
