5.2 LOGARITHMIC FUNCTIONS & THEIR GRAPHS
Goals—Recognize and evaluate logarithmic functions with base aGraph Logarithmic functionsRecognize, evaluate, and graph natural logsUse logarithmic functions to model and solve real-life problems.
Is this function one
to one?
Horizontal Line test?
Does it have an inverse?
f(x) = 3x
LOGARITHMIC FUNCTION WITH BASE “A”
Definition
For x > 0, a > 0, and a 1,y = logax if and only if x =
ay
The function given by f(x) = logax read as “log base a of x”
is called the logarithmic function with base a.
WRITING THE LOGARITHMIC EQUATION IN EXPONENTIAL FORM
log381 = 4 log168 = 3/4
82 = 64 4-3 = 1/64
34 = 81 163/4 = 8
log 8 64 = 2 log4 (1/64) = -3
WRITING AN EXPONENTIAL EQUATION IN LOGARITHMIC FORM
EVALUATING LOGSy= log232
y= log42
y= log31
y= log101/100
Step 1: Rewrite the log problem as an exponential.
Step 2: Rewrite both sides of the = with the same base.
2y = 32
2y = 25
Therefore, y = 5
y = 5
4y = 2(22)y = 21
22y = 21
y = 1/2y = 1/2
3y = 1y = 0
y = 0 10y = 1/100
y = -210y = 10-2
y = -2
EVALUATING LOGS ON A CALCULATOR
f(x) = log x when x = 10 f(x) = 1 when x = 1/3 f(x) = -.4771 when x = 2.5 f(x) = .3979 when x = -2 f(x) = ERROR!!! Why???
You can only use a calculator when the base is 10
PROPERTIES OF LOGARITHMS
loga1 = 0 because a0 = 1
logaa = 1 because a1 = a
logaax = x and alogax = x
logax = logay, then x = y
SIMPLIFY USING THE PROPERTIES OF LOGS
log41
log77
6log620
Rewrite as an exponent4y = 1 So y = 0
Rewrite as an exponent7y = 7 So y = 1
USE THE 1-1 PROPERTY TO SOLVE
log3x = log312
log3(2x + 1) = log3x
log4(x2 - 6) = log4 10
x = 12
2x + 1 = xx = -1
x2 - 6 = 10x2 = 16x = 4
F(X) = 3X
Graphs of Logarithmic Functions
So, the inverse would be
g(x) = log3x
Make a T chart
Domain—Range?
Asymptotes?
Graphs of Logarithmic Functions
g(x) = log4(x – 3)
Make a T chart
Domain—Range?
Asymptotes?
Graphs of Logarithmic Functionsg(x) = log5(x – 1) +
4
Make a T chart
Domain—Range?
Asymptotes?
NATURAL LOGARITHMIC FUNCTIONS
The function defined by f(x) = loge x = ln x, x > 0
is called the natural logarithmic function.
EVALUATEf(x) = ln x when x = 2 f(x) = .6931 when x = -1 f(x) = Error!!! Why???
PROPERTIES OF NATURAL LOGARITHMS
ln 1 = 0 because e0 = 1
ln e = 1 because e1 = e
ln ex = x and elnx = x (Think…they are inverses of each other.)
If ln x = ln y, then x = y
USE PROPERTIES OF NATURAL LOGS TO SIMPLIFY EACH EXPRESSION
ln (1/e) = ln e-1 = -1 eln 5 = 5 2 ln e = 2
Graphs of Natural Logs
g(x) = ln(x + 2)
Make a T chart
Domain—Range?
Asymptotes?
2 Undefined 3 4
Graphs of Natural Logsg(x) = ln(2 - x)
Make a T chart
Domain—Range?
Asymptotes?
2 Undefined 1 0
