1 6.4 logarithmic functions in this section, we will study the following topics: evaluating...
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6.4 Logarithmic Functions
In this section, we will study the following topics:
Evaluating logarithmic functions with base a
Graphing logarithmic functions with base a
Evaluating and graphing the natural logarithmic function
Solving logarithmic and exponential equations
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Logarithmic Functions
Now that you have studied the exponential function, it is time to take
a look at its INVERSE: THE LOGARITHMIC FUNCTION.
In the exponential function, the independent variable (x) was the
exponent. So we substituted values into the exponent and
evaluated it for a given base.
Exponential Function: f(x) = 2x, f(3) = 23 = 8.
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Logarithmic Functions
For the inverse function (LOGARITHMIC FUNCTION), the base is
given and the answer is given, so to evaluate a logarithmic
function is to find the exponent.
That is why I think of the logarithmic function as the “Guess That
Exponent” function.
Warm Up: Give the value of ? in each of the following equations.
? ? ?11) 3 81 2) 5 3) 16 4
25
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Exponential and logarithmic functions of the same base are inverses.
Exponential and logarithmic functions of the same base are inverses.
Exponential and logarithmic functions of the same base are inverses.
Exponential and logarithmic functions of the same base are inverses.
Exponential and logarithmic functions of the same base are inverses.
Exponential and logarithmic functions of the same base are inverses.
Exponential and logarithmic functions of the same base are inverses.
Exponential and logarithmic functions of the same base are inverses.
Exponential and logarithmic functions of the same base are inverses.
Subliminal Message
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Logarithmic Functions (continued)
Evaluate log28
To evaluate log28 means to find the exponent such that 2 raised to
that power gives you 8.
?
2log 8 ?
2 8
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Logarithmic Functions (continued)
The following definition demonstrates this connection between the exponential and the logarithmic function.
Definition of a Logarithmic FunctionFor x > 0, a > 0, and a ≠ 1,
y = logax if and only if x = ay
We read logax as “log base a of x”.
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Converting Between Exponential and Logarithmic Forms
I. Write the logarithmic equation in exponential form.
a)
b)
II. Write the exponential equation in logarithmic form.
a)
b)
3log 81 4
7
1log 2
49
329 27
2 18
64
y = logax if and only if x = ay
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Evaluating Logarithms w/o a Calculator
To evaluate logarithmic expressions by hand, we can use the related
exponential expression.
Example:
Evaluate the following logarithms:
10 5
110,000 b)) log log
25a
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Evaluating Logarithms w/o a Calculator (cont.)
336) 6 d) log 1logc
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Evaluating Logarithms w/o a Calculator
Okay, try these.
e) f)
g) h)
5log 5 4log 0
8
1log
2 10log 0.0001
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The Common Logarithm
The common logarithm has a base of 10. If the base
of a logarithm is not indicated, then it is assumed
that the base is 10.
10For example, log 0.01 is equivalent to log 0.01
log if and only if 10 yy x x
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Graphs of Logarithmic Functions
Since the logarithmic function is the _______________ of the
exponential function (with the same base), we can use what
we know about inverse functions to graph it.
Example:
Graph f(x) = 2x and g(x) = log2x in the same coordinate plane.
To do this, we will make a table of values for f(x)=2x and then
switch the x and y coordinates to make a table of values for g(x).
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Graphs of Logarithmic Functions (continued)
f(x) = 2x g(x) = log2x
x f(x)
-4
-2
0
2
4
x g(x)
f(x) = 2x
g(x)= log2x
y =x
Inverse functions
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Comparing the Graphs of Exponential and Log Functions
Notice that the domain and range of the inverse functions are switched.
The exponential function has
domain (-, )
range (0, )
HORIZONTAL asymptote y = 0
The logarithmic function has
domain (0, )
Range (-, )
VERTICAL asymptote x = 0
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Transformations of Graphs of Logarithmic Functions
The same transformations we studied earlier also apply to logarithmic functions. Look at the following shifts and reflections of the graph of f(x) = log2x.
The new vertical asymptote is x = -2
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Transformations of Graphs of Logarithmic Functions
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The Natural Logarithmic Function
In section 6.3, we saw the natural exponential function with base
e. Its inverse is the natural logarithmic function with base e.
Instead of writing the natural log as logex, we use the
notation , which is read as “the natural log of x” and is
understood to have base e.
ln x
ln if and only if yy x x e
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Natural Log Key
To evaluate the natural log using the TI-83/84, use the button.
Notice, the 2nd function of this key is ________.
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Graph of the Natural Exponential and Natural Logarithmic Function
f(x) = ex and g(x) = ln x are inverse functions and, as such, their graphs
are reflections of one another in the line y = x.
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Evaluate without using a calculator.
a) b)
c) d)
e) f)
3ln e
5ln eln ( 2)
4ln e
1lne
ln1
Evaluating the Natural Log
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Solving Logarithmic Equations
Strategy for solving logarithmic equations:
Change the equation from a log equation into an exponential equation, using one of the following forms: logax = y x = ay
logx = y x = 10y
lnx = y x = ey
Keep in mind that the domain of the log function is x>0. Reject any extraneous solutions!!
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Examples of solving log equations
51) log 3x 32) log 3 2 2x
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More examples of solving log equations
63) log 36 5 3x 254) log 4 2x x
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End of Section 6.4