Warmup
Solve for x by guess and check.
1. 5x = 25
2. 4x = 1
3. 3x = 1/27
4. ex = e6
1
3.2 Logarithmic Functions
Definition of a logarithm:
y = loga x if and only if x = ay
*Important Note: x > 0 and a > 0
Function Notation:
f(x) = loga x
(base a)
To evaluate a logarithm ask the following question:
"a to what power is going to give me x?"
2
Converting logarithms to exponents
1. log2 16 = 4
*Remember 2 is the base of the logarithm and is the base of the
exponential expression.
2. log9 1/81 = -2
3. ln e4 = 4
Converting exponents to logarithms
4. 72 = 49
5. 2-3 = 1/8
6. ex = 4
3
Evaluate the following:
1. log2 32
(2 to what power gives me 32?)
2. log3 27
3. log4 2
4. log71
4
*Now you try:
5. log10 1000
6. log10 1/100
7. logaa3
8.
log48
5
The Natural Logarithm
f(x) = loge x = ln x
Evaluate:
1. ln 1/e
2. ln e5
3. 2 ln e
6
The log function with base 10 is called the
common logarithmic function. On most
calculators, this is denoted by LOG .
So, we can evaluate log functions with base
10 approximately using calculators.
7
Use your calculators to evaluate.
Round your answer to 3 decimals if necessary.
1. log 10
2. log 3.2
3. log -5
4. 3 ln 7
5. ln (1/2)
8
Change of base formula (used for plugging
logs of any base into calculators)
log a b =
log b
log a
or
ln b
ln a
Example:
log 2 5
log37
9
Steps for graphing a logarithmic function
y = c logb(x ­ h) + k
Initial Point: (1, 0) or (­1, 0)
only use if -x
h:
k:
if c is negative:
if x is negative:
asymptote:
domain:
range:
10
Sketch a graph of the following:
1. y = ln (x + 2) - 3
11
2. y = - log2x - 4
12
3. y = log (3 - x) + 2
13
4. y = - ln (-x - 4) - 1
14
Homework:
P. 236
#3, 6, 9, 12, 16, 19-21,
31, 37, 53, 56
P. 244
#3, 4, 7
15