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Goals—•Recognize and evaluate logarithmic functions with base a•Graph Logarithmic functions•Recognize, evaluate, and graph natural logs•Use logarithmic functions to model and solve real-life problems.
Is this function one
to one?
Does it have an inverse?
Must pass the horizontal line
test.
Yes
Yes
f(x) = logax is called the logarithmic function of base a.
Definition: Logarithmic function of base “a” -
For x > 0, a > 0, and a 1,y = logax if and only if x = ay
Read as “log base a of x”
The most important thing to remember about logarithms
is…
a logarithm is an exponent.
Therefore, all logarithms can be written as exponential equations and all exponential equations can be written as logarithmic equations.
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
f(x) = log42
f(x) = log31
f(x) = log10(1/100)
Step 1- rewrite it as an exponential
equation. 2y = 32Step 2- make the bases
the same. 2y = 25
Therefore,
y = 5
4y = 222y = 21
y = 1/2
3y = 1y = 0
10y = 1/100
10y = 10-2
y = -2
Think: y = log232
f(x) = log232
You can only use a calculator when the base is10
Find the log key on your calculator.
Evaluate the following using that log key.
log 10 = 1
log 1/3 = -.4771
log 2.5 = .3979
log -2 = ERROR!!!
Why?
loga1 = 0 because a0 = 1logaa = 1 because a1 = alogaax = x and alogax = xIf logax = logay, then x = y
log41=
log77 =
6log620 =
Rewrite as an exponent 4y = 1 Therefore, y = 0
Rewrite as an exponent 7y = 7 Therefore, y = 1
0
1
20
log3x = log312
log3(2x + 1) = log3x
log4(x2 - 6) = log4 10
x = 12
2x + 1 = xx = -1
x2 - 6 = 10x2 = 16x = 4
Review: How do you find the inverse of a function?
Application of what you know…What is the inverse of f(x) = 3x?
y = 3x
x = 3y
y = log3xf-1(x) = log3x
Rewrite the exponential as a logarithm…
Find the inverse of the following exponential functions…
f(x) = 2x f-1(x) = log2x f(x) = 2x+1 f-1(x) = log2x - 1
f(x) = 3x- 1 f-1(x) = log3(x + 1)
Find the inverse of the following logarithmic functions…
f(x) = log4x f-1(x) = 4x
f(x) = log2(x - 3) f-1(x) = 2x + 3
f(x) = log3x – 6 f-1(x) = 3x+6
Graphs of Logarithmic Functions
It is the inverse of y = 3x
y = 3x
x y-1 1/3
0 11 32 9
y= log3x
x y1/3 -1
1 03 19 2
Therefore, the table of values for
g(x) will be the reverse of the
table of values for y = 3x.
Domain? (0,)
Range? (-,)
Asymptotes? x = 0
Graphs of Logarithmic Functionsg(x) = log4(x – 3)What is the inverse exponential function?
y= 4x + 3Show your tables of values.
y= 4x + 3x y-1 3.250 41 72 19
y= log4(x – 3)
x y3.25 -1
4 07 1
19 2
Domain? (3,)
Range? (-,)
Asymptotes? x = 3
Graphs of Logarithmic Functionsg(x) = log5(x – 1) + 4What is the inverse exponential function?
y= 5x-4 + 1Show your tables of values.
y= 5x-4 + 1
x y3 1.24 25 66 26
y= log5(x – 1) + 4
x y1.2 32 46 5
26 6
Domain? (1,)
Range? (-,)
Asymptotes? x = 1
The function defined by f(x) = logex = ln x, x > 0
is called the natural logarithmic function.
Find the ln key on your calculator.
Evaluate the following using that ln key.
ln 2 = .6931
ln 7/8 = -.1335ln 10.3 = 2.3321ln -1 = ERROR!!!
Why?
ln1 = 0 because e0 = 1Ln e = 1 because e1 = eln ex = x and eln x = xIf ln x = ln y, then x = y
ln 1/e=
2 ln e =
eln 5=
Rewrite as an exponent ey = 1/e
ey = e-1
Therefore, y = -1Rewrite as an exponent ln e = y/2
e y/2 = e1 Therefore, y/2 = 1 and
y = 2.
-1
2
5
Graphs of Natural Log Functionsg(x) = ln(x + 2)
Show your table of values.
y= ln(x + 2)
x y-2 error-1 00 .6931 1.0992 1.386
Domain? (-2,)
Range? (-,)
Asymptotes? x = -2
Graphs of Natural Log Functionsg(x) = ln(2 - x)
Show your table of values.
y= ln(2 - x)
x y2 error1 00 .693-1 1.099-2 1.386
Domain? (-2,)
Range? (-,)
Asymptotes? x = -2