Date post: | 31-Dec-2015 |
Category: |
Documents |
Upload: | erasmus-lopez |
View: | 51 times |
Download: | 3 times |
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.
Logarithmic function with base “b”
The logarithm to the base “b” of a positive number y is defined as follows:
If y = bx , then logby = x
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
Evaluating Logs
f(x) = log232 f(x) = log42
f(x) = log31
f(x) = log10(1/100)
Step 1- rewrite it as an exponential equation.
Step 2- make the bases the same.
Common Logarithms
• Use base 10
• Written –• Log10y is the same as log y
• Only Common Logs can be evaluated using a calculator.
Evaluating Logs on a Calculator
f(x) = log x when x = 10 when x = 1/3 when x = 2.5 when x = -2
You can only use a calculator when the base is 10
f(x) = 1
f(x) = -.4771
f(x) = .3979
f(x) = ERROR!!! Why?
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
Now graphg(x) = log3x
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.
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