8.4 Notes day 1 (simplifying)
Logarithmic Functions
In 8.1-8.3, we solved story problems involving the formulas:
“The dark side calls constantly for
aggression, revenge, betrayal. The stronger
you become, the more you're tempted."
What do we do if we want to know how much time it will take for the Principal to reach the given Amount?
More specifically, we can simplify this question to “What is x if 10
x
40 ?
Define:
Example 1:
Evaluating logarithmic expressions
a) log 3 81
b) log4 64
d) log5 0.04
e)
c) log5 625
log 1 8
f) log9 3
2
g)
log 4 16 x
h)
log 9 81 x
i) log 3 81
There are two special logarithms that we will encounter:
j) log 10000
k)
log .01
2x 4
l)
ln e 20 x
x 4
If the logarithm “undoes” the exponential base, then an exponential must “undo” a logarithm as well…
Example 2:
a)
Simplify the following:
5 log 5 x
b) 6
log 6 ( 2 x 4 )
c)
10 log(17
42 x 2 )
d) e
ln(12345)
If two functions “undo” each other, we call them:
Let
Thus
g ( x)
log b x and let f ( x) b x
f ( g ( x) )
and
If we go back to the example we started with, 10
x
g ( f ( x) )
40 , then how do we solve this for x?
Example 3: Find the inverses of the logarithmic and exponential functions.
a)
y
log 3 x
b) y
log 2 x
5
c)
y
log 4 ( x 6) 3
d)
y
log(x 9) 14
e)
y
ln(x 2) 7
f)
y
2 3x
g)
y
h)
y
4
10 x
3
4 ex
5
9
1
6
12
Because exponential equations and logarithmic equations are inverses of each other, it is possible to quickly switch between forms.
Example 4:
Quickly switch between the two forms.
logarithmic form
a)
logarithmic form
log2 32 5
e)
50
b)
c)
exponential form
1
g)
10
1
log1/ 2 2
01
.
9
log4 1 0
h)
Some special logarithms (NEED TO MEMORIZE): If b is a positive real number other than 1
Logarithm of one :
1
91
f)
log10 10 1
d)
exp. form
Logarithm of same number as base:
34
81