Logarithmic

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Logarithmic Functions
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The logarithmic function to the base a, where a gt
0 and a ? 1 is defined
y logax if and only if x a y
logarithmic form
exponential form
When you convert an exponential to log form,
notice that the exponent in the exponential
becomes what the log is equal to.
Convert to log form
Convert to exponential form
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LOGS EXPONENTS
With this in mind, we can answer questions about
the log
This is asking for an exponent. What exponent do
you put on the base of 2 to get 16? (2 to the
what is 16?)
What exponent do you put on the base of 3 to get
1/9? (hint think negative)
What exponent do you put on the base of 4 to get
1?
When working with logs, re-write any radicals as
rational exponents.
What exponent do you put on the base of 3 to get
3 to the 1/2? (hint think rational)
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Logs and exponentials are inverse functions of
each other so lets see what we can tell about
the graphs of logs based on what we learned about
the graphs of exponentials.
Recall that for functions and their inverses, xs
and ys trade places. So anything that was true
about xs or the domain of a function, will be
true about ys or the range of the inverse
function and vice versa.
Lets look at the characteristics of the graphs
of exponentials then and see what this tells us
about the graphs of their inverse functions which
are logarithms.
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Characteristics about the Graph of an Exponential
Function a gt 1
Characteristics about the Graph of a Log Function
where a gt 1
1. Domain is all real numbers
1. Range is all real numbers
2. Range is positive real numbers
2. Domain is positive real numbers
3. There are no x intercepts because there is no
x value that you can put in the function to make
it 0
3. There are no y intercepts
4. The x intercept is always (1,0) (xs and ys
trade places)
4. The y intercept is always (0,1) because a 0
1
5. The graph is always increasing
5. The graph is always increasing
6. The x-axis (where y 0) is a horizontal
asymptote for x ? - ?
6. The y-axis (where x 0) is a vertical
asymptote
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Logarithmic Graph
Exponential Graph
Graphs of inverse functions are reflected about
the line y x
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Transformation of functions apply to log
functions just like they apply to all other
functions so lets try a couple.
up 2
Reflect about x axis
left 1
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Remember our natural base e? We can use that
base on a log.
What exponent do you put on e to get 2.7182828?
ln
Since the log with this base occurs in nature
frequently, it is called the natural log and is
abbreviated ln.
Your calculator knows how to find natural logs.
Locate the ln button on your calculator. Notice
that it is the same key that has ex above it.
The calculator lists functions and inverses using
the same key but one of them needing the 2nd (or
inv) button.
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Another commonly used base is base 10.A log to
this base is called a common log.Since it is
common, if we don't write in the base on a log it
is understood to be base 10.
What exponent do you put on 10 to get 100?
What exponent do you put on 10 to get 1/1000?
This common log is used for things like the
richter scale for earthquakes and decibles for
sound.
Your calculator knows how to find common logs.
Locate the log button on your calculator. Notice
that it is the same key that has 10x above it.
Again, the calculator lists functions and
inverses using the same key but one of them
needing the 2nd (or inv) button.
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The secret to solving log equations is to
re-write the log equation in exponential form and
then solve.
Convert this to exponential form
check
This is true since 23 8
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Acknowledgement I wish to thank Shawna Haider
from Salt Lake Community College, Utah USA for
her hard work in creating this PowerPoint. www.sl
cc.edu Shawna has kindly given permission for
this resource to be downloaded from
www.mathxtc.com and for it to be modified to suit
the Western Australian Mathematics Curriculum.
Stephen Corcoran Head of Mathematics St
Stephens School Carramar www.ststephens.wa.edu.
au