10.4 Properties of Logarithms - Snow College

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Transcript of 10.4 Properties of Logarithms - Snow College

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