# 10.4 Properties of Logarithms - Snow College

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2.7 Further Applications of Linear Equations• Use properties to
write alternative forms of logarithmic

expressions.

Product Rule for Logarithms

If , , and are positive real numbers, where ≠ 1,

then the following is true.

That is, the logarithm of a product is the sum of the logarithms of the factors.

Using the Product Rule

Assume > .

log4 3 7

Assume > .

log5 6 9

Quotient Rule for Logarithms

If , , and are positive real numbers, where ≠ 1,

then the following is true.

That is, the logarithm of a quotient is the difference between the

logarithm of the numerator and the logarithm of the denominator.

log7 9

log3 − log3

• Use the quotient rule to rewrite each logarithm. Assume > .

Using the Quotient Rule

Assume > .

Assume > .

5

Power Rule for Logarithms

If , , and are positive real numbers, where ≠ 1, and

if is any real number, then the following is true.

That is, the logarithm of a number to a power equals the

exponent times the logarithm of the number.

log5 2 3

log2 7 4

Using the Power Rule

Assume > , > , > and ≠

log3 5 2

Special Properties

If > 0 and ≠ 1, then the following are true.

Using the Special Properties

5log5 9

log2 8

4log4

Properties of Logarithms If x, y, and b are positive real numbers, where ≠ 0, and r is any

real number, then

x x y

log logr

5 5 5log 6 9 log 6 log 9

4 4 4

9

2

4log 10 4

54 10 log 5 4and

Writing Logarithms in Alternative Forms Use the properties of logarithms to rewrite each expression of

possible. Assume that all variables represent positive real

numbers.

Use the properties of logarithms to rewrite each expression if

possible. Assume that all variables represent positive real

numbers.

Use the properties of logarithms to rewrite each expression if

possible. Assume that all variables represent positive real

numbers.

log 82

− 1

Writing Logarithms in Alternative Forms Use the properties of logarithms to rewrite each expression as a

single logarithm if possible. Assume that all variables

represent positive real numbers.

4 log − log

Use the properties of logarithms to rewrite each expression as a

single logarithm if possible. Assume that all variables

represent positive real numbers.

2 log + 3 log

Use the properties of logarithms to rewrite each expression as a

single logarithm if possible. Assume that all variables

represent positive real numbers.

3 log

Numerical Values

Given that log = . and log = . , evaluate the

following.

log2 15 log2 27

Given that log = . and log = . , evaluate the

following.

Decide whether each statement is true or false.

log2 8 − log2 4 = log2 4

Decide whether each statement is true or false.

log4 log2 16 = log6 6

log6 36

log3 log2 8 = log7 49

log8 64

expressions.

Product Rule for Logarithms

If , , and are positive real numbers, where ≠ 1,

then the following is true.

That is, the logarithm of a product is the sum of the logarithms of the factors.

Using the Product Rule

Assume > .

log4 3 7

Assume > .

log5 6 9

Quotient Rule for Logarithms

If , , and are positive real numbers, where ≠ 1,

then the following is true.

That is, the logarithm of a quotient is the difference between the

logarithm of the numerator and the logarithm of the denominator.

log7 9

log3 − log3

• Use the quotient rule to rewrite each logarithm. Assume > .

Using the Quotient Rule

Assume > .

Assume > .

5

Power Rule for Logarithms

If , , and are positive real numbers, where ≠ 1, and

if is any real number, then the following is true.

That is, the logarithm of a number to a power equals the

exponent times the logarithm of the number.

log5 2 3

log2 7 4

Using the Power Rule

Assume > , > , > and ≠

log3 5 2

Special Properties

If > 0 and ≠ 1, then the following are true.

Using the Special Properties

5log5 9

log2 8

4log4

Properties of Logarithms If x, y, and b are positive real numbers, where ≠ 0, and r is any

real number, then

x x y

log logr

5 5 5log 6 9 log 6 log 9

4 4 4

9

2

4log 10 4

54 10 log 5 4and

Writing Logarithms in Alternative Forms Use the properties of logarithms to rewrite each expression of

possible. Assume that all variables represent positive real

numbers.

Use the properties of logarithms to rewrite each expression if

possible. Assume that all variables represent positive real

numbers.

Use the properties of logarithms to rewrite each expression if

possible. Assume that all variables represent positive real

numbers.

log 82

− 1

Writing Logarithms in Alternative Forms Use the properties of logarithms to rewrite each expression as a

single logarithm if possible. Assume that all variables

represent positive real numbers.

4 log − log

Use the properties of logarithms to rewrite each expression as a

single logarithm if possible. Assume that all variables

represent positive real numbers.

2 log + 3 log

Use the properties of logarithms to rewrite each expression as a

single logarithm if possible. Assume that all variables

represent positive real numbers.

3 log

Numerical Values

Given that log = . and log = . , evaluate the

following.

log2 15 log2 27

Given that log = . and log = . , evaluate the

following.

Decide whether each statement is true or false.

log2 8 − log2 4 = log2 4

Decide whether each statement is true or false.

log4 log2 16 = log6 6

log6 36

log3 log2 8 = log7 49

log8 64