5.2 Logarithmic Functions & Their Graphs
Goals—Recognize and evaluate logarithmic functions with base a
Graph Logarithmic functionsRecognize, evaluate, and graph natural logs
Use logarithmic functions to model and solve real-life problems.
f(x) = 3x
Is this function one to one?
Horizontal Line test?
Does it have an inverse?
Logarithmic function with base “a”
• Def’n of Logarithmic function with base “a”
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.
Write the logarithmic equation in exponential form
log381 = 4 log168 = 3/4
Write the exponential equation in logarithmic form
82 = 64 4-3 = 1/64
34 = 81 163/4 = 8
log 8 64 = 2 log4 (1/64) = -3
Evaluating Logs
f(x) = log232 f(x) = log42
f(x) = log31
f(x) = log10(1/100)
Step 1- rewrite it as an exponential equation. 2y = 32
Step 2- make the bases the same.
2y = 25
Therefore, y = 5
4y = 222y = 22y = 2y = 1
3y = 1y = 0
10y = 1/10010y = 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 beg(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 Functions
g(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
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