Lesson 5.3: Properties of Logarithms Change-of-Base Formulas Let - - PowerPoint PPT Presentation
Lesson 5.3: Properties of Logarithms Change-of-Base Formulas Let a, b, and x be positive real numbers such that a 1 and b 1. Then log a x can be converted to a different base as follows. Base 10 Base b Base e log x log x ln x
log log log
a x
x a
10 10
log log log
a b b
x x a
Change-of-Base Formulas Let a, b, and x be positive real numbers such that a 1 and b 1. Then loga x can be converted to a different base as follows. Base 10 Base b Base e
log ln ln
a x
x a
Ex 1: Rewrite each log as a ratio of common logs and natural logs.
A.
C.
log log
10 10
30 4
ln ln 30 4
log log
10 10
17 2
ln ln 17 2
log log
10 10 6
x
ln ln x 6
Let a be a positive number such that a 1, and let n be a real number. If u and v are positive real numbers, the following properties are true.
Logs with base a 1.Loga (uv) = loga u + loga v 2.Loga u/v = loga u – loga v
Need to be the same!
Natural Logs
Let a be a positive number such that a 1, and let n be a real number. If u and v are positive real numbers, the following properties are true.
Ex 2: Write each logarithm in terms of ln 2 and ln 3
F
ln 2 27 F
Ex 3: Expand each logarithmic expression.
B.
ln 3 5 7 x
4 4 3 4
4 4 4
1 2
Ex 4: Condense each logarithmic expression.
A.
1 2 3 1
10 10
log log x x
10 1 2 10 3
10 10 3
3
Ex 4: Condense each logarithmic expression.
2 ln ln x x
2
2
Homework: p.393 #3-15, 39-78 all mult. of 3 C.
Ex 4: Condense each logarithmic expression.
2 2
2
1 3
3