Transcript Revised Logarithmic Functions

Logarithmic
Functions
The logarithmic function to the base a, where a > 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:
16  4
2
log 416  2
Convert to exponential form:
1
log 2  3
8
2
3
1

8
LOGS = EXPONENTS
With this in mind, we can answer questions about the log:
log 2 16  4
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?)
1
log 3  2
9
What exponent do you put on the base of
3 to get 1/9? (hint: think negative)
log 4 1  0
1
2
1
log33 33 
log
2
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)
 1, x  0
BASIC PROPERTIES OF
LOGARITHMS
For 0 < b 1, x > 0, and any real number y,
log b 1  0 because b = 1.
0
log
bb 1
because b
1
b
y
b
y
log b b  y
because b
logb x= x
b
because log b x  log b x
y
BASIC PROPERTIES OF
COMMON LOGARITHMS
Let x and y be real numbers with x > 0
log 1 = 0
0
because 10 = 1
1
log 10 = 1 1 because 10 =10
o
y
y
log 10 = y
10
log x
0
y
because 10 = 10
1
=x
because log x = log x
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.
log 100  2
1
log
 3
1000
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.
Basic Properties of Natural
Logarithms
Let x and y be real numbers with x > 0
0
0
ln 1 = 0 because e =1
1
ln e = 1 because e = e
y
y
ln e = y because e = e y
e ln x = x because ln x = ln x
y
1
Remember our natural base “e”?
We can use that base on a log.
exponent do you put
log e 2.7182828  1 What
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.
ln 2.7182828  1
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.
The secret to solving log equations is to re-write the
log equation in exponential form and then solve.
log 2 2x  1  3
2  2x 1
3
8  2x  1
7  2x
7
x
2
Convert this to exponential form
check:
 7 
log 2  2   1  3
 2 
log 2 8  3
This is true since 23 = 8
In the last section we learned about the graphs of
exponentials.
Logs and exponentials are inverse functions of each other
so let’s 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, x’s and y’s
trade places. So anything that was true about x’s or
the domain of a function, will be true about y’s or the
range of the inverse function and vice versa.
Let’s 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.
Characteristics about the
Graph of an Exponential
Function f x   a x a > 1
Characteristics about the
Graph of a Log Function
f x   log a x where a > 1
1. Domain is all real numbers 1. Range is all real numbers
2. Range 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
4. The y intercept is always
(0,1) because a 0 = 1
5. The graph is always
increasing
6. The x-axis (where y = 0) is
a horizontal asymptote for
x-
2. Domain is positive real
numbers
3. There are no y intercepts
4. The x intercept is always
(1,0) (x’s and y’s trade places)
5. The graph is always
increasing
6. The y-axis (where x = 0) is
a vertical asymptote
Exponential Graph
Graphs of
inverse
functions are
reflected about
the line y = x
Logarithmic Graph
Transformation of functions apply
to log functions just like they apply
to all other functions so let’s try a
couple.
up 2
f x   log 10 x
f x   2  log 10 x
Reflect about x axis
f x    log 10 x
left 1
f x   log 10 x  1