Download Day III 3.3 Notes

Math Analysis
9/1/11
Day III. Properties of Logarithms (3.3)
"The greater part of our happiness or misery depends on our dispositions, and not on our
circumstances." Martha Dandridge Custis Washington, 1731 – 1802
Logarithmic functions are often used to model scientific observations like human memory.
GOAL I. To rewrite logarithmic functions with a different base
I.
Change of Base
Calculators only have two types of log keys. The common log and the natural log. The bases are 10
and e, respectively.
Change-of-Base Formula
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 b
Base 10
Base e
loga x =
loga x =
loga x =
Example 1. Changing Bases Using Common Logarithms
Using a calculator and the common log setting, evaluate the expression to 1/10000.
log7 4 =
Your Turn
1. log1/4 5 =
2. log20 0.125 =
Example 2. Changing Bases Using Natural Logarithms
Using a calculator and the natural log setting, evaluate the expression to 1/10000.
log7 4 =
Your Turn
1. log1/4 5 =
2. log20 0.125 =
GOAL II. To use properties of logarithms to evaluate or rewrite logarithmic expressions
II. Properties of Logarithms
Summative Math Algebra 2 Standard 14.0.1 - Students understand the properties of logarithms
(log laws).
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.
u
= loga u – loga v
v
log u = log u – log v
v
u
ln
= ln u – ln v
v
1. loga(uv) = loga u + loga v
2. loga
log (uv) = log u + log v
ln (uv) = ln u + ln v
3. loga un = n loga u
log un = n log u
ln un = n ln u
Example 3. Using Properties of Logarithms
Use the properties of logarithms and the given values to find the logarithm indicated. NO
CALCULATORS!!!
log 7  0.8
log 8  0.9
log 12  1.1
1.
log
7
=
8
2.
log 64 =
3.
log 96 =
Your Turn
1. log
7
=
12
2.
log 49 =
3.
log 1008 =
Example 4. Using Properties of Logarithms
Use the properties of logarithms and the given values to find the logarithm indicated. NO
CALCULATORS!!!
1
=
16
1.
log9 7 = A
log9 4 = B
log9 10 = C
log9
3.
log8 12 = P
log8 5 = Q
log8 9 = R
log8 32
27
2.
=
log7 6 = R
log7 8 = S
log7 10 = T
log7 392 =
Your Turn
1.
log5 12 = R
log5 9 = S
log5 11 = T
log5
3.
log7 3 = X
log7 8 = Y
log7 10 = Z
log7
1
=
12
2.
log8 6 = A
log8 9 = B
log8 10 = C
log8 729 =
15
=
32
GOAL III. To use properties of logarithms to expand or condensed logarithmic expressions
III. Rewriting Logarithmic Expressions
Summative Math Algebra 2 Standard 14.0.3 - Students use the properties of logarithms to identify their
approximate values (expanding).
Example 5.
Expanding Logarithmic Expressions
Use properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of
logarithms. Assume all variables are positive.
1.
x2 - 1
x3
ln
,x>1
2.
ln x2(x + 2)
Your Turn
1.
ln
x
=
x2 + 1
2.
ln
x2
y3
Summative Math Algebra 2 Standard 14.0.2 - Students use the properties of logarithms to simplify
logarithmic numeric expressions (condensing).
Example 6. Condensing Logarithmic Expressions
Condense the expression to the logarithmic of a single quantity.
1.
4[lnz + ln(z + 2)] – 2ln(z – 5)
Your Turn
1.
2ln 8 + 5ln z
2.
2[lnx – ln(x + 1) – ln(x – 1)]
GOAL IV. To use logarithmic functions to model and solve real-life applications
IV. Applications
Logarithmic functions are often used to model scientific observations like human memory.
Example 7. Finding a Mathematical Model
Students participating in a psychological experiment attended several lectures and were given an
exam. Every month for a year after the exam, the students were retested to see how much of the
material they remembered. The average score of the group can be modeled by the memory model
f(t) = 90 – 15 log (t + 1), 0  t  12 where t is the time in months.
1. What was the average score on the original exam (t = 0)?
2. What was the average score after six months?
Your Turn
3. What was the average score after 12 months?
4. When will the average score decrease to 75?