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Advanced Algebra
Properties of Logarithms
Name:______________________________
Date:__________________ Per._________
Product Property of Logarithms: log=
log b m + log b n
b mn
If m, n, and b are positive real numbers and
b ≠ 1.
Example: log 9 10 + log 9 3 =
log 9 30
Write as a single logarithm.
1
2. log 6 + log 6 15 = 3. log 3 x + log 3 ( x − 9 ) =
3
m
Quotient Property of Logarithms: log
If m, n, and b are positive real numbers
=
log b m − log b n
b
n
and b ≠ 1 .
Example: log 3 45 − log 3 5 =
log 3 9
1. log 2 7 + log 2 5 =
Write as a single logarithm.
4. log 7 40 − log 7 8 =
5. log 9 x − log 9 5 =
6. log 3 ( x 2 + 4 ) − log 3 ( x 2 + 2 ) =
Power Property of Logarithms:
log b m p = p log b m
If m and b are positive real numbers, b ≠ 1 ,
and p is a real number.
1
log 4 2
2
#7-8: Use the power property to rewrite.
8. log 7 3 4 =
7. log 3 x5 =
1/ 2
Example: log
=
2 log
=
4
4 2
Exponential-Logarithmic Inverse Properties:
#9: Write as a single logarithm.
9. 3log 4 2 + 2 log 4 5 =
log b b x = x and b logb x = x for x > 0
If b > 0 and b ≠ 1
Example: 7 log7 4 − log 3 81 = 4 − log 3 34 = 4 − 4 = 0
#10-11: Evaluate the expression
10. log8 85 + 3log3 8 =
11. log 3 35 + log 5 125 =
One-to-One Properties of Logarithms:
If log b x = log b y , then x = y
Example: log 4 ( 3 x=
− 1) log 4 ( 2 x + 8 ) then 3x – 1 = 2x + 8 so x = 9
#12-13: Solve for x and check your answer.
12. log8 ( x=
+ 1) log8 ( 2 x − 2 )
log b x =
Change-of-Base Formula:
Example: Evaluate log8 36
14. Evaluate. log 1 15 =
2
log
=
8 36
13. 2 log b x =
log b 2 + log b ( 2 x − 2 )
log a x
log a b
For any positive real numbers a ≠ 1 , b ≠ 1 , x > 0
log 36
≈ 1.73 By changing the base to 10, you can use a calculator.
log 8
15. Solve. 2.5 x = 17
Advanced Algebra
Properties of Logarithms
Name:______________________________
Date:__________________ Per._________
Product Property of Logarithms: log=
log b m + log b n
b mn
If m, n, and b are positive real numbers and
b ≠ 1.
Example: log 9 10 + log 9 3 =
log 9 30
Write as a single logarithm.
1
2. log 6 + log 6 15 =
log 6 5 3. log 3 x + log 3 ( x −=
9 ) log 3 ( x 2 − 9 x )
3
m
Quotient Property of Logarithms: log
If m, n, and b are positive real numbers
=
log b m − log b n
b
n
and b ≠ 1 .
Example: log 3 45 − log 3 5 =
log 3 9
1. log 2 7 + log 2 5 =
log 2 35
Write as a single logarithm.
6. log 3 ( x + 4 ) − log 3 ( x + 2 ) =
log 3
x
5. log 9 x − log 9 5 =
log 9
5
4. log 7 40 − log 7 8 =
log 7 5
2
log b m p = p log b m
Power Property of Logarithms:
2
(x
(x
2
2
+ 4)
+ 2)
If m and b are positive real numbers, b ≠ 1 ,
and p is a real number.
1
log 4 2
2
#7-8: Use the power property to rewrite.
1/ 2
Example: log
=
2 log
=
4
4 2
#9: Write as a single logarithm.
1
8. log 7 3 4 = log 7 4
3
7. log 3 x5 = 5log 3 x
Exponential-Logarithmic Inverse Properties:
9. 3log 4 2 + 2 log 4 5 =
log 4 200
log b b x = x and b logb x = x for x > 0
If b > 0 and b ≠ 1
Example: 7 log7 4 − log 3 81 = 4 − log 3 34 = 4 − 4 = 0
#10-11: Evaluate the expression
10. log8 85 + 3log3 8 =
13
11. log 3 35 + log 5 125 =
8
One-to-One Properties of Logarithms:
If log b x = log b y , then x = y
Example: log 4 ( 3 x=
− 1) log 4 ( 2 x + 8 ) then 3x – 1 = 2x + 8 so x = 9
#12-13: Solve for x and check your answer.
log8 ( x=
+ 1) log8 ( 2 x − 2 )
12.
x=3
log b x =
Change-of-Base Formula:
Example: Evaluate log8 36
#14: Evaluate.
14. log 1 15 = −3.90689
2
log
=
8 36
13.
log a x
log a b
2 log b x =
log b 2 + log b ( 2 x − 2 )
x=2
For any positive real numbers a ≠ 1 , b ≠ 1 , x > 0
log 36
≈ 1.73 By changing the base to 10, you can use a calculator.
log 8
#15: Solve.
2.5 x = 17
15.
x = 3.9205