Download 14.1 Exponential Functions and Applications 14.3

14.1 Exponential Functions and
Applications
14.3 Logarithmic Functions and
Applications
14.3 Logarithmic Functions
• Logarithms have many applications inside and
outside mathematics. Some of these occurrences
are related to the notion of scale invariance. For
example, each chamber of the shell of a nautilus
is an approximate copy of the next one, scaled by
a constant factor
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•To find the number of payments on a loan or the time to reach an investment goal
•To model many natural processes, particularly in living systems. We perceive loudness of
sound as the logarithm of the actual sound intensity, and dB (decibels) are a logarithmic
scale. We also perceive brightness of light as the logarithm of the actual light energy, and
star magnitudes are measured on a logarithmic scale.
•To measure the pH or acidity of a chemical solution. The pH is the negative logarithm of
the concentration of free hydrogen ions.
•To measure earthquake intensity on the Richter scale.
•To analyze exponential processes. Because the log function is the inverse of the
exponential function, we often analyze an exponential curve by means of logarithms.
Plotting a set of measured points on “log-log” or “semi-log” paper can reveal such
relationships easily. Applications include cooling of a dead body, growth of bacteria, and
decay of a radioactive isotopes. The spread of an epidemic in a population often follows a
modified logarithmic curve called a “logistic”.
•To solve some forms of area problems in calculus. (The area under the curve 1/x,
between x=1 and x=A, equals ln A.)
• Other uses that are more aligned with the scope
of this course are determining decibel levels for
sound (loudness) and severity of earthquakes.
Logarithms are used in other areas but the math
needed to apply them is advanced beyond this
class. We will use a few of these concepts to
learn how to deal with logs through that use
hopefully you learn an appreciation for the need
of logs.
First of all, what is a logarithm? You’ve worked with them before, you’ve learned the rules
before, but do you really know what one is?
• In its simplest form, a logarithm answers the question:
• How many of one number do we multiply to get another
number?
For example how many 2s do you need to multiply together
to get 8?
The log then would be 3.
The notation would be
log 2 8  3
How many of the bases “2” do you
need to multiply by itself to get 8?
Remember logs and exponentials are
inverses of one another.
• Take your calculator, find the log button, the log
button on your calculator uses a base of 10. Later
we can change that, but for right now our
calculator is only useful when the base is 10.
log10 6.3  0.8
• Verify the above value with your calculator. What does
this mean.
– To get a value of 6.3, you need to multiply 10 by itself 0.8 times.
Common Log
• A log that involves a base of 10 is referred to as the
common log. The definition is as follows ….
log x  a iff 10  x
a
General Log Rule for logs of bases
other than 10
log b x  a iff x  b
a
log10 1000  3
log 2 32  5
log10 0.1  1
1
log16 4 
2
Write the following in log form
103 = 1,000 42 = 16 33 = 27 51 = 5 70 = 1 4-2 = 1/16 251/2 = 5
x  b iff log b x  a
a
Examples of Logarithms
Exponential Form
Logarithmic Form
23  8
1 4
 2   16
51  5
3 0
4   1
Example
Solution
log 2 8  3
log 16  4
log 5 5  1
log 1  0
1
3
Solve log 4 x  32 .
log 4 x  32
x4
x 8
3
2
2
4
Solving Logarithmic Equations
Example
Solve a) x  log 8 4 b) log x 16  4.
Solution
a) x  log 8 4
8x  4
2 
3 x
 22
23 x  2 2
3x  2
2
x
3
b) log x 16  4
4
x  16
x   4 16
x  2
Since the base must
be positive, x = 2.
The Common Logarithm – Base 10
For all positive numbers x,
log x  log 10 x.
Example
Evaluate a) log12 b) log 0.1 c) log 53 .
Solution
Use a calculator.
a) log 12  1.079181246
b) log 0.1  1
3
c) log  .2218487496
5
The Natural Logarithm – Base e
For all positive numbers x,
ln x  log e x.
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On the calculator, the natural logarithm key is
usually found in conjunction with the e x key.