L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
Recall that an exponential function is a one-to-one function that has the form f  x   a x .
where a > 0 and
a≠1
a is the BASE and is a constant.
x is the exponent and a variable.
This gave us a family of curves, each member of which had a different base.
We used f  x   2x as an example because it is easy to graph.
page 34
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
What is inverse of f  x   2x ? That is, what is f
Graph of f
1
1
page 35
 x  ? We discussed this last lecture:
 x  is a reflection of f(x) about the line y
 x . So, we can draw it.
We don’t know what the inverse function is and we cannot solve this equation for y.
So, what do mathematicians do when they cannot solve something? Yes, they make up something!
The inverse of the exponential function is so important that it is given a special name: The Logarithm.
Here is how we define logarithm today:
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
page 36
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
page 37
From the definition we can see that a logarithm is an exponent.
This definition allows us to switch between exponential and logarithmic functions.
log5 125  3

Check: f  f
other.
1
can be written as
 x   5
log5  3 
53  125
 3 The exponential operating on the log of the same base undo each
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
page 38
We can find the values of some logs exactly by converting them to exponentials.
This is the power of 2 that yields
1
8
Set the log equal to a variable and then convert to an exponential.
Let y  log2
1
8
then
2y 
1
8
Now, all we have to do is solve this exponential, which we know how to do.
1
8
1
2y  3
2
y
2  23
2y 
Since the expressions are equal and the bases are equal, the exponents must be equal.
So, y = -3.
We could also solve this by noting that log functions and exponential functions are inverses
and the composition of inverse functions "cancel" the functions. That is,
f
f
1
  x   f f  x    x
1
1
1
 log2 3  log2 23 . We can think of this as f  x   log2 x and therefore
8
2
1
x
f  x   2 . Thus we have a log base 2 operating on its inverse, an exponential base 2, so they
We can write log2
cancel each other to give -3.
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
2 yields
That is, what power of
1
?
16
2 .
Convert this to an exponential, base
Let y  log
2
1
16

then
2

y

1
16
Now, solve the exponential equation by writing each side with the same base.

y
2 
y

2
12

1
16
 2 4
y
2 2  2 4
y
2
 4
y  8  log
1
16
2
OR
We could solve this by writing 1/16 as a power of
1
1
 4 
16 2
So, log
2
1
2
1
•8
2

1
2 
1
 log
16
1/ 2
2
8

 21/ 2
 
2
8

8

 8
 2
8
2  21/ 2
page 39
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
page 40
The restriction on the base of a log is that it must be positive and not 1. This gives us an infinite
number of bases to use. There are two that are used most often.
Base 10: These are called common logs. If you don’t write a base, a base of 10 is assumed. This is
because the base of our number system is 10. So, log x  log10 x
n

1
Base e: Recall that e is Euler's number and is defined as e  lim  1   = 2.7182818284590...
n 
n

Base e logs are called natural logs since the number e occurs in all sorts of natural places,
especially in growth and decay problems. We abbreviate these logs as ln.
So, ln x  loge x
There are two logs keys on your calculator: one is labeled log (base 10) and the other ln (base e). If
you need to calculate a log of a different base you can use either log or ln and a formula called the
change of base formula (covered next lecture).
http://en.wikipedia.org/wiki/Logarithmic_spiral#Logarithmic_spirals_in_nature
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
page 41
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
page 42
Logarithms can also be defined with a Taylor series: http://www.math.com/tables/expansion/log.htm
http://kr.cs.ait.ac.th/~radok/math/mat6/calc6.htm
To graph log functions we can often use our knowledge of transformations as we have done for lots
of other functions.
Let’s do an example (http://en.wikipedia.org/wiki/Carbon_dating)
Cosmic rays in the upper atmosphere are constantly converting the isotope nitrogen-14 (N-14) into
carbon-14 (C-14 or radiocarbon). Living organisms are constantly incorporating this C-14 into their
bodies along with other carbon isotopes. When the organisms die, they stop incorporating new C-14,
and the old C-14 starts to decay back into N-14 by beta decay (emits an electron from its nucleus).
The older an organism's remains are, the less beta radiation it emits because its C-14 is steadily
dwindling at a predictable rate. So, if we measure the rate of beta decay in an organic sample, we
can calculate how old the sample is. C-14 decays with a half-life of 5,730 years. Archeologists use
this information to help them date ancient materials. (This, of course, assumes that the rate of decay
of C-14 has been constant over time and that it is not affected by its environment and that the relative
concentrations of C-12 and C-14 have remained constant).
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
page 43
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
page 44
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
page 45
First, we find the decay constant for c-14. We can do this knowing the half-life of c-14.
A t   A0e kt
1
k 5730 
A0  A0e 
2
1
k  5730 
e
2
 1
k  5730 
ln    ln e
2


 1
ln    5730k
2
k  0.000121
Now we know the decay constant so we can write: A t   A0e 0.000121t
Since 1.67% of the original amount is left since the tree died, the current amount is 0.0167 Ao
0.0167A0  A0e 0.000121t
0.0167  e 0.000121t
ln  0.0167   0.000121t
t  33, 821
We estimate that the tree died about 33,800 years ago.
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
Now, let's solve a few more logarithmic equations.
Convert to an exponential and solve:
x

log2   3   2
4

x
4
x
 3  2 2
1
4
4
x  12  1
3
x  13
Check: See if x = 13 works.
x

log2   3   2
4

 13

log2 
 3   2
 4

 13 12 
log2 

 2
4 
 4
 1
log2    2
4
log2 22  2
2  2
page 46
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
We know how to solve a radical and absolute value equation of this form:
32
2

3  2 2  x   1
2  x   1
 2  x   2
2  x   1
2  x  
2
 1
2  2  x   2
2  x   1
2x 1
x 1
2
or
or
2x 1
x 1
Solve for log term, convert to an exponential, and solve:
3  2 log  2  x   1
3  2 log  2  x   1

log  2  x   1
101  2  x
x  8

3  2 log 2   8   1
2 log  2  x   2
Check:
3  2 log10  1
321
1 1
2  x  1
x 3
page 47
L34-Mon-21-Nov-2016-Sec-5-4-Log-Functions-HW35-Moodle-Q28
We can use logs to solve exponential equations also.
Solve for the e term and then use logs.
3  1  e 2 x 1
2  e 2 x 1
ln 2  2x  1
ln 2  1
x
2
0.69  1
x
2
0.85  x
3  1 e
2  0.85   1
0.70
Check: 3  1  e
3  1  2.01
3  3.01
page 48