Transcript No Slide Title

Common Logarithms
• If x is a positive number, log x is the exponent of 10 that gives x.
That is, y = log x if and only if 10y = x.
• The function log x is known as the common logarithm function
or, simply, the log function. It may also be written as log10 x,
which is read as “log to the base 10 of x”.
• Example.
is log 100? Since 102 = 100, we have log 100 = 2.
What
What
is log 0.01? Since 10-2 = 0.01, we have log 0.01 = -2.
Since
100.5  10 ,
log 10 ? we have
What is
log 10  0.5.
• Two important values of the log function are:
log 1  0
and
log 10  1.
• By definition, the log is the inverse function of the
exponential with base 10, so we have:
log10 x  x, for all x,
10log x  x, for x  0.
• For a and b both positive and any value of t,
log (ab)  log a  log b
a
log    log a  log b
b
log(b t )  t  log b .
How many years will it take for your salary to double?
• Problem. If you start at $40000, and you are given a 6%
raise each year, how many years must pass before your
salary is at least $80000?
Solution. We must solve (1.06)t(40000) = 80000 for t.
Equivalently, we must solve (1.06)t = 2 for t. If we take
the log of both sides of this equation and use the last
property on the previous slide, we obtain
t  log 1.06  log 2, or
log 2
0.30103
t

 11.896 years.
log 1.06 0.025306
If you have to wait until the end of the year to actually get
your raise, 12 years must pass.
Half-life of the dotcom investment
• How many years must pass before the investment discussed
previously has a value which is one-half of its original
value? We must solve the following equation for t:
(0.95) t (10000)  5000, which becomes
(0.95) t  0.5, upon dividingby 10000.
t  log 0.95  log 0.5 is obtained after taki ng logs.
log 0.5
 0.30103
t

 13.51 years.
log 0.95  0.022276
Graphical Solution to dotcom half-life
(13.51, 5000)
Solving a logarithmic equation
• Solve for x:
log(2x+1) = 3
10log(2x+1) = 103
exponentiate
2x+1 = 1000
evaluate
2x = 999
subtract
x = 499.5
divide
Fallacies Involving Logs
• log(a + b) is not the same as log a + log b .
• log(a – b) is not the same as log a – log b .
• log(ab) is not the same as (log a)(log b) .
log a
a
• log  is not the same as
.
log b
b
1
1
• log  is not the same as
.
log a
a
Natural Logarithms
• If x is a positive number, ln x is the exponent of e that
gives x. That is, y = ln x if and only if ey = x.
• The function ln x is known as the natural logarithm
function. It may also be written as loge x, which is read as
“log to the base e of x”.
• Example.
is ln e? Since e1 = e, it follows that ln e = 1.
What
What is ln
(1/e)? Since e-1 = 1/e, it follows that ln (1/e) = -1.
What is
ln e  0.5.
e 0.5 thate ,
Since ln e ? it follows
• Two important values of the ln function are:
ln 1  0
and
ln e  1.
• By definition, ln is the inverse function of the exponential
with base e, so we have:
ln e x  x, for all x,
e ln x  x, for x  0.
• For a and b both positive and any value of t,
ln (ab)  ln a  ln b
a
ln    ln a  ln b
b
ln b t  t  ln b .
Converting Between Q = abt and Q = aekt
• Any exponential function can be written in either of two
forms: Q = abt and aekt.
If b  e k , then k  ln b.
• Problem. Convert the exponential function P = 175(1.145)t
to the form P = aekt.
Solution. We must find k such that
e k  1.145, or
k  ln 1.145  0.1354.
Therefore,
P  175e0.1354t .
Logarithms and exponential models
• Problem 47, Section 4.2. Suppose the temperature H, in °F,
of a cup of coffee t hours after it is set out to cool is given
by the equation:
H  70  120(1/4)t .
How long does it take the coffee to cool down to 90°F?
Solution. We must solve the following equation for t:
70  120(1/4)t  90,
120(1/4)t  20, by subtracting
(1/4) t  1/6, by dividing
log(1/4)t  log(1/6), by takinglogs
t  log(1/4)  log(1/6), using a log property
t  log(1/6)/log(1/4)  1.29 hours.
Some problems require graphical methods for solution
• Problem. Find the positive value of x which satisfies:
10 x  x  2.
If we take the log of both sides, we get an equation with a
logarithm term and we are unable to isolate x.
Solution. Use graphical approach:
The relation between ln x and log x
• We start with the equation
10log x  x
and we apply ln to both sides of this equation. We have
ln 10 log x  ln x.
Next, we apply a property from a previous slide to get
log x  ln 10  ln x.
The last equation tells us that ln x equals log x times the
constant factor, ln 10 = 2.3026. Because of this fact, the
shapes of the graphs of ln x and log x are similar.
The graph, domain, and range of the common logarithm
• It follows from the definition of log x that its domain consists
of all positive real numbers. Its range is all real numbers.
x
log x
0.01
-2
0.1
-1
1
0
10
1
100
2
1000
3
• Using Maple or graphing calculator, we can plot the graph of
log x:
The graphs of 10x and log x using Maple
> plot({10^x,x,piecewise(x>0,log10(x))},x=-4..10,-4..10,color=black,
scaling=constrained);
More Detail for Graphs of 10x and log(x)
· (0.3010,2)
(0,1)
(2,0.3010)
·
(–1,0.1)
·
(1,0)
· (0.1,–1)
Asymptote for log x
• The exponential function 10x has the property that its graph
approaches the x-axis as x gets large and negative. We
recall that this behavior is described by saying that 10x has
the x-axis as a horizontal asymptote. We can also express
this by writing 10x  0 as x   or by lim 10 x  0.
x  
• Corresponding to the above property for the exponential is
the property that the graph of log x approaches the negative
y-axis as x approaches 0 from the right. This behavior is
described by saying that log x has the negative y-axis as a
vertical asymptote. We can also express this by writing
log x   as x  0 or by lim log x   .
x0
Chemical Acidity
• In chemistry, the acidity of a liquid is expressed using pH.
The acidity depends on the hydrogen ion concentration in
the liquid (in moles per liter). This concentration is written
[H+]. The pH is defined as:
pH   log [H  ].
• Problem. A vinegar solution has a pH of 3. Determine the
hydrogen ion concentration.
Solution. Since 3 = – log[H+], we have –3 = log[H+]. This
means that 10-3 = [H+]. The hydrogen ion concentration is
10-3 moles per liter.
Logarithms and orders of magnitude
• We often compare sizes or quantities by computing their
ratios. If A is twice as tall as B, then
Height of A/Height of B = 2.
• If one object is 10 times heavier than another, we say it is
an order of magnitude heavier. If one quantity is two
factors of 10 greater than another, we say it is two orders
of magnitude greater, and so on.
• Example. The value of a dollar is two orders of magnitude
greater than the value of a penny.
$1
 10 2 .
$0.01
We note that the order of magnitude is the logarithm of the
ratio of their values.
Decibels
• To measure a sound in decibels, the sound’s intensity, I, in
watts/cm2 is compared to a standard benchmark sound, I0.
This results in the following definition:
 I 
,
Noise level in decibels  10  log
 I0 
where I0 is defined to be 10-16 watts/cm2, roughly the
lowest intensity audible to humans.
• Problem. If a sound doubles in intensity, by how many
units does its decibel rating increase?
 2I 
 I


Difference in decibel ratings  10  log
 10  log

 I0 
 I0
  2I 
 I 

  log    10  log2  3.010 dB.
 10 log


I
I
0
0









Summary for Logarithmic Functions
• log x is the exponent of 10 that gives x. The log function is the
inverse function of the exponential function with base 10.
• ln x is the exponent of e that gives x. The ln function is the
inverse function of the exponential function with base e.
• log x and ln x have several “properties of logarithms”.
• Using logarithms, we can solve certain equations involving
exponential functions.
• The domain of the logarithm is all positive numbers, and the
range of the logarithm is all real numbers.
• The graphs of the logarithm and the exponential function (with
the same base) are “mirror images” across the line y = x.
• The graph of the logarithm has the y-axis as a vertical asymptote.
• Logarithms are used to measure quantities which vary over a wide
range. E. g., hydrogen ion concentration & sound intensity.