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Overview Study Material Business Mathematics 1 2015-2016
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1
Algebra
The Real Numbers
The basic numbers are 1,2,3,4…, these numbers are natural numbers and also called
positive integers. The positive integers, together with the negative integers 0,-1,-2,-3,…,
make up the integers. The integers can be represented on a number line (showed
below), where the arrow provides the direction in which the numbers increase.
Rational numbers are all numbers that are written in the form c/d (where c and d are
both integers). They can also be presented on a number line. An integer (say n) is also a
rational number, because n = n/1. There are also Irrational Numbers like √3, this is
because there are no integers p and q such that √3 = p/q.
The way people write numbers today is called the decimal system or base 10 system.
This is a positional system with 10 as the base number, where every natural number can
be written by using only the symbols: 1,2,3,4 (…) 9, which are called digits. An example
of how the decimal system works:
This enables us to also express rational numbers other than natural numbers, for
example:
Rational numbers that can be written exactly, by using only a finite number of decimal
places, are called finite decimal fractions. If this is not the case, we call them infinite
decimal fractions. An example of an infinite decimal fraction is: 20/6 = 3,3333…, where
the three dots indicate that the digit 3 is repeated indefinitely. This is also called a real
number.
When applied to a real number, the four arithmetic operations (addition, subtraction,
multiplication and division) always result in a real number. The only exception is that
they cannot devide by 0.
Integer Powers
When b is any number and n is any natural number, the following applies:
This expression is called the nth power of b: where b is the base and n the exponent.
A special case is , where b can be any number except 0. Powers with negative
exponents can be defined as:
0123 1071021031011327
43210 10/510/310/710/21042735,4
bn = b×b×... ×bn factors
10 b
nn bb /1
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Properties of Powers
There are several rules that apply to Powers. The most important two are:
The division of two powers with the same base goes like:
Finally, note that:
One of the most common errors committed is:
is NOT (in general).
Compound Interest
Suppose a company deposits an amount of K euros in a bank account paying p% of
interest per year. After t years the amount of money in the account will be:
Where is the growth factor for a growth of p%.
If you want to calculate the original amount of money deposited in the bank, t years ago.
Where the bank pays an interest rate of p% per year, use the following formula, where K
is the amount of money now and X is the original amount:
This formula can also be used for calculating the depreciation of an asset. Suppose a car
worth K euros, with a depreciation rate of p%. After t years, the value of the car will be:
Here is called the growth factor for a decline of p%.
Rules of Algebra
If a, b, and c are arbitrary numbers, the following rules are the most important:
These rules can be combined.
lklk aaa lklk aa )(
lklk aaa /
kkk cbbc )(kkkkk cbcbcb /)/(
kcb )( bk +ck
tpK )
1001(
1001
p
tpKX )
1001/(
tpK )
1001(
1001
p
abba )()( cbacba
aa 00)( aa
baab )()( bcacab
aa 1
011 avooraa
abbaba )()(
abba ))((
acabcba )(
bcaccba )(
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3
Quadratic Identities
There are 3 important quadratic identities, which are very important to remember:
The last one is called differences-of-squares-formula. This is the proof for the formula:
Algebraic Expressions
The expression is an algebraic expression, because it involves
letters. We call the terms in the expression. The numbers 5, 3 and 7
are the numerical coefficients of the terms. Terms where only the numerical coefficient is
different, are called terms of the same type. For example:
Factoring
When we write 81 as 9x9, we have factored this number. Algebraic expressions can be
factored in a similar way:
With Factoring, we mean: to express it as a product of simpler factors. However, most
algebraic expressions can not be factored.
Fractions
In the equation , c is the numerator and d is the denominator. The fraction is a
proper fraction because the numerator is smaller than the denominator. An improper
fraction is: , because the numerator is larger (or equal to) the denominator. This
fraction can be re-written as: which is called a mixed number.
222 2)( bababa 222 2)( bababa
22))(( bababa
2222))(( babababababa
322 735 abaab
5ab,3a2and7b2a3
4242 63 zxenzx
)3(103010 2 xxxx
d
c
7
4
10
15
2
11
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The most important properties of fractions are listed below:
Fractional Powers
An example of a power with a fractional exponent is , when and x = ½. We
define as √b, the square root of b.
√a is a nonnegative number because when multiplied by itself, the answer gives a.
If a and b are nonnegative numbers, then two rules apply:
(i)
(ii)
For the second rule, b ≠ 0.
In addition:
It is known that and have the same solution, which is 4. When solving the
equation , the solution must be written like . Note however, that √4
means only 2, not -2.
Nth Roots
The following rule applies to :
We can expand this rule:
Here a must be positive (a>0), in addition p must be an integer and q must be a natural
number.
0,0
cb
b
a
cb
ca
b
a
b
a
b
a
)1()(
)1()(
b
a
b
a
b
a
b
a
)1()1(
c
ba
c
b
c
a
db
cbda
d
c
b
a
c
bca
c
ba
c
ba
c
ba
db
ca
d
c
b
a
cb
da
c
d
b
a
d
c
b
a
xb 0b2/1b
b
a
b
a
baba
22 22
42 x 24 x
a1/n
nn aa /1
pqpqqp aaa )()( /1/
baab
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Inequalities
Properties:
a>0 and b>0, imply a+b>0 and a x b > 0;
a>b means that a-b>0;
a≥b means that a-b≥0;
If a > b, then a + c > b + c for all c.
To deal with more complicated inequalities involves using the following properties:
According to these properties, the following rules apply:
If the two sides of the inequality are multiplied by a positive number, the direction
of the inequality is preserved;
If the two sides of an inequality are multiplied by a negative number, the direction
of the inequality is reversed.
Sign Diagrams
For the solution of an equation, for example , a sign diagram might be
useful. For an example of a sign diagram, review the compulsory study material.
Double Inequalities
An example of a double inequality is:
It is natural to write:
Intervals and Absolute Values
Let y and z be any two numbers on the real line. All the numbers that lie between y and
z, are called an interval. There are four different intervals that all have a and b as
endpoints (see the figure below).
Notation Name Interval x
(a, b) Open interval
[a, b] Closed interval
(a,b] Half-open interval
[a,b) Half-open interval
All four intervals have the same length (b –a), only the endpoints differ. The open
interval for example uses neither of the endpoints. All the intervals mentioned above are
bounded intervals. An example of an unbounded interval is:
[a,∞) = for all values of x, with
Here, ∞ stands for infinity. This symbol is not a number at all, the symbol only indicates
that we are considering the collection of all numbers larger than a, without any upper
bound on the size of the number. Similarly, (-∞,b) has no lower bound. When all real
numbers are permitted, we say: (-∞, ∞).
a> 0 and b> 0; a+b> 0; a×b> 0
a> bthen a+c> b+c
a> b and b> c®a> c
a> b and c> 0®ac> bc
a> b and c< 0®ac< bc
a> b and c> d®a+c> b+d
0)2)(5( xx
bdendc
bdc
bxa
bxa
bxa
bxa
ax
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Absolute value
The absolute value of b is denoted |b|, and:
Suppose two numbers, a en b. The distance between a and b on the number line is a-b if
and –(a-b) if a<b. Therefore, we have the distance between a and b on the
number line defined as:
Absolute values can also be part of an inequality:
b = bif b³ 0 and -bif b< 0{ }
ba
abba
x < a means-a< x < a
x £ a means-a£ x £ a
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A. Equations
How to solve simple equations
To solve an equation means to find all values of the variables for which the equation is
satisfied. If any value of a variable makes an expression in an equation undefined, that
value is not allowed. An example is: z/(z-5), here z can not be 5 because it makes the
expression undefined, because it is 5/0.
When two equations have exactly the same solution, we call them equivalent. To get
equivalent equations, you have to do the following on both sides of the equality sign (=):
Add or subtract the same number;
Multiply or divide by the same number (≠0).
Equations with Parameters
These equations have a common structure, which makes it possible to write down a
general equation covering all the special cases:
Here a and b are real numbers, also called parameters. The variables in the equation
are x and y.
Consider the basic macroeconomic model:
Where Y is the GDP (Gross Domestic Product), C is consumption, and is total
investment (which is treated as fixed). Here, a and b are the positive parameters, with
b<1. These equations represent the structural form of the model.
Substituting C = a + bY + I gives us:
Next, rearrange the equation so that all the terms containting Y are on the left-hand
side:
The left-hand side is equal to Y(1-b), so devide both sides by 1-b so that the coefficient Y
becomes 1. This gives us the answer:
This is one part of the reduced form, that expresses endogenous variables as
functions of exogenous variables.
Quadratic Equations
The general quadratic equation (also called second-degree equation) has the form:
Where x is the unknown variable, and a, b, and c are given constants.
baxy
ICY
bYaC
I
IbYaY
IabYY
I
bb
aY
1
1
1
)0(02 acbxax
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The Quadratic formula
To solve a quadratic equation like: , it is possible to use the quadratic
formula.
If: Then: If and only if:
When < 0, the square root of a negative number appears and no real solution
exists. The solutions of the equation are often called the roots of the equation.
Because the Quadratic formula contains a ± sign, there are two solutions, and ,
that make sure the quadratic equation equals 0. This can be written as:
then:
Linear Equations in Two Unknowns
This section will review some methods for solving two linear equations with two
unknowns:
We must find the values of x and y that satisfy both equations. There are two methods
for this:
Method 1
(1) Solve one of the equations for one of the variables in terms of the other, then (2)
substitute the result into the other equation:
1)
2)
The solutions should always be checked by direct substitution.
Method 2
This method is based on eliminating one of the variables by adding or subtracting a
multiple of one equation from the other. In the following equation, we will eliminate x.
We multiply the second equation with -2/3, then we will add the transformed equations.
The term x disappears and we obtain:
-------------------------
Hence, y = 4 which gives us x=3 (through substitution of 4 for y). To solve a general
linear system, we use the following two equations and two unknowns:
02 cbxax
0042 aacb 02 cbxaxa
acbbx
2
42
b2 - 4ac
1x 2x
0))(( 21 xxxx ))(( 21
2 xxxxacbxax
3664 yx
1486 yx
xy 4366
xy 3/26
14)3/26(86 xx
143/16486 xx343/111 x
43 yx
3664 yx
-4x+5 1/ 3y= 9 1/ 3
3664 yx
3/193/154 yx
3/1453/1110 y
cbyax
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Here a, b, c, d, e, and f are arbitrary given numbers, whereas x and y are the unknowns.
Using Method 2, we multiply the first equation by e and the second by –b to obtain the
following, which gives the value for x:
By substituting it back into or
, we get y.
Nonlinear Equations
An example of a nonlinear equation is: . People tend to calculate the solution
in the following way:
However, this is an easy way to make a serious mistake, because here a factor is
cancelled which might be zero (x=0 is a solution too!).
A safer method is . When the equation is in this form, it is easier to see that
x=0 is also a solution.
In general:
is equivalent to a=0 of b=c.
feydx
)/()( bdaebfcex
)/()( bdaecdafy
cbyax feydx
06 23 xx
23 6xx
6x
0)6(2 xx
acab
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B. Summation
The capital Greek letter ∑ is used as a summation symbol, and the sum is written as:
This reads ‘the sum from i=1 to n of ’. If there are n regions, then:
The symbol i is called the index of summation. It is a ‘dummy variable’ that can be
replaced by any other letter (which has not already been used for something else).
The lower and upper limits of the summation can both vary, for example:
The solution to the first summation:
In order to summarize the overall effect of price changes, a number of price indices
have been suggested. A price index for year t, in which year 0 as the base year, is
defined as:
Here q(i) is the number of goods (in the basket), p0(i) is the price per unit of good i in
year 0, and pt(i) is the price per unit of good i in year t.
In the case where the quantities are levels of consumption in the base year 0, this is
called: the Laspeyres price index. But if the quantities are levels of consumption
in year t, this is called the Paasche price index.
A few aspects of logic
Propositions are assertions that are either true or false. This is an example of a
mathematical proposition: .
Implications
Suppose P and Q are two propositions, such that whenever P is true, then Q is
necessarily true. If this is the case, we usually write:
The arrow is an implication arrow and it points in the direction of the logical
implication.
n
i
iN1
iN
nNNN ...21
r
qi
i
i
beni5
1
2
5554321 222225
1
2 i
i
100
1
)()(
0
1
)()(
n
i
ii
n
i
ii
t
qp
qp
)(iq)(iq
042 x
BA
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In some cases, where the previous implication is valid, it may also be possible to draw a
logical conclusion in the other direction:
In such cases, we can write both implications together in a single logical equivalence,
as shown below:
The symbol is called an equivalence arrow.
Necessary and Sufficient Conditions
There are other ways of expressing that proposition P implies proposition Q, or that P is
equivalent to Q.
To emphasize this point, consider the following two propositions:
Breathing is a necessary condition for a person to be healthy;
Breathing is a sufficient condition for a person to be healthy.
The first proposition is obviously true. The second proposition is false, because a sick
person is still breathing.
BA
AÛB
Û
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C. Functions of one variable
One variable is a function of another if the first variable depends upon the second, for
instance . Here, when x is given, y can be determined, y is a function of x. In this
situation x can be any number. However, mostly there are some restrictions like: only
the values 1 to 10 are relevant. Sometimes a graph is preferable to a formula, for a
better overview of the situation.
Basic Definitions
A function of real variable x with domain is a rule that assigns a unique real number
to each real number x in D. As x varies over the whole domain (D) the set of all possible
values f(x) is called the range of . Functions are given letter names like f, z, g, F, or
φ.
If f is a function, we often let y denote the value of f at x, so:
We call x the independent variable or the argument of f, whereas y is the dependent
variable, because the value y depends on the value of x (in general). The domain of f is
then the set of all possible values of the independent variable, whereas the range is the
set of corresponding values of the dependent variable. In economics, x is called the
exogenous variable, which is supposed to be fixed OUTSIDE the economic model,
whereas for each given x the equation y = f(x) serves to determined the endogenous
variable y INSIDE the economic model.
Functional Notation
Consider the following function:
Now suppose all values of x increase with 1. A common mistake is:
The right answer is:
Substituting a for x in the formula of f gives:
Whereas (if a a + 1):
Domain and Range
The definition of a function is not really complete unless its domain is either specified
explicitly or obvious. So: if a function is defined using an algebraic formula, the domain
consists of all values of the independent variable for which the formula gives a unique
value (unless another domain is explicitly mentioned).
xy 2
fD
fR
)(xfy
2)( xxf
1)1( 2 xxf
2)1()1( xxf
2)( aaf
1212)1)(1()1()()1( 22222 aaaaaaaaaafaf
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For example, we are going to find the domain of:
For x=2, the formula reduces to a meaningless expression (6/0). For all other values of
x, the formula makes f(x) a well-defined number. Thus, the domain consists of all
numbers except x = 2.
Another example, find the domain:
The function is uniquely defined for all x such that 6 +x is nonnegative. Thus, the domain
of the function can be written as the interval: [-6,∞). The range of the function can also
be calculated. The range of f is all the numbers we get as output, using all numbers in
the domain as imputs. The range of the function is in this case[0,∞).
A function f is called increasing if implies . A function is strictly
increasing if implies . Decreasing and strictly decreasing are
defined in the opposite way.
Graphs of Functions
A rectangular or Cartesian coordinate system is obtained by first drawing two
perpendicular lines, called coordinate axes. The two axes are respectively the y-axis (the
vertical axis) and the x-axis (the horizontal axis). The intersection point O is called the
orgin. This rectangular coordinate system is also called the x y-plane. We measure the
real numbers along of each line, although the unit distance on the x-axis is not
necessarily the same as on the y-axis.
Any point P in the plane can be represented by a unique pair of real numbers (c,d). The
point represented by (a, b) lies at the intersection of x=c and y=d. We call (c, d) the
coordinates of P, (c, d) is also called an ordered pair because the order of the two
numbers in the pair is important. For instance, (4, 5) and (5, 4) represent two
completely different points.
Each function of one variable can be represented by a graph in such a rectangular
coordinate system. The graph of a function f is simply the visual representation of the set
of all points (x, f(x) ), where x belongs to the domain of f.
Linear Functions
Linear functions occur a lot in economics, and they are defined as:
where a and b are constants.
The graph of this equation is a straight line. Suppose x increases with 1, then:
This shows that a measures the change in the value of the function when x increases by
1 unit, for this reason a is the slope of the line or the function.
x
xxf
2
3)(
)6( x
21 xx )()( 21 xfxf
21 xx )()( 21 xfxf
baxy
abaxbxaxfxf )1()()1(
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If the slope of a is positive, the line goes upward to the right, and the larger the value of
a, the steeper is the line. On the other hand, if the value of a is negative, then the line
slants downward to the right, and again the absolute value of a measures the steepness of the line. For example, when a is -6 the steepness is 6.
In the special case, when a (the slope) equals 0, the steepness is zero because the line is
horizontal. Algebraically, we have y = ax + b =b for all x. Here, b is called the y-
intercept.
The Point-Slope and Point-Point Formulas
Suppose point S(x1, y1). Let us find the straight line equation passing through point P
with slope a. If is any other point on the line, then the slope a is given by the
formula:
where
Graphical solutions of Linear Equations
Earlier, we have dealt with algebraic methods for solving a system of two linear
equations in two unknowns. The equations are linear, so naturally their graphs are
straight lines. The coordinates of any point on the line satisfy the equation of that line.
Thus, the coordinates of any point of the intersection of these two lines will satisfy both
equations. This means that any point where these lines intersect will satisfy the equation
system.
Linear inequalities
Suppose the linear inequality . The inequality can be written as y ≤ 8 - 4x. The
set of points (x, y) that satisfy the inequality y ≤ 8 - 4x must have y-values below those
points on the line y = -4x + 8, because y = -4x +8 is a straight line.
Linear Models
In Keynesian macroeconomic theory, total consumption expenditure (on goods and
services) is C. C is assumed to be a function of national income Y:
This is called the Consumption Function.
The slope of b is called the marginal propensity to consume. This tells us by how
many consumption changes if national income increases with 1. C, Y, and b can also be
measured in billions of dollars for example.
The point where demand is equal to supply, represents an equilibrium. The price P* at
which this occurs is the equilibrium price and the corresponding quantity Q* is the
equilibrium quantity.
Consider the following simple example of demand and supply functions:
Equilibrium occurs at: P*= 36 and Q*= 164.
These are the general linear demand and supply schedules:
Here a, b, , and are the positive parameters of the functions.
),( 22 yxT
)(
)(
12
12
xx
yya
21 xx
84 yx
bYaCYfC )(
PD 200
PS 420
bPaD
PaS
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The formula for the equilibrium Price is:
The formula for the equilibrium Quantity is:
Quadratic Functions
The general quadratic function is:
Where a, b, and c are constants and a ≠ 0.
In general, the graph of is called a parabola.
The shape of a parabola roughly resembles an upside down U when a < 0, and a U when
a > 0. Parabolas are symmetric about the axis of symmetry, for examples review the
compulsory literature.
To calculate for which value of x, f(x) is equal to 0, you can use the quadratic formula.
Suppose a≠0 and then:
Where stands for the two possible values where y=0.
Furthermore:
To find the maximum/minimum of the function use the following formulas:
When a>0, has a minimum of at .
When a<0, has a
maximum of at .
Polynomials
Polynomials or cubic functions have a general
form:
Here a, b, c, and d are constants and a≠0. On
the right is an example of a cubic function.
General Polynomials
Cubic, quadratic and linear functions are all examples of polynomials. The function P
defined for all x by:
b
aP
*
b
baQ
*
cbxaxxf 2)(
cbxaxxf 2)(
042 acb
a
acbbx
2
42
2,1
2,1x
))(( 21
2 xxxxacbxax
cbxaxxf 2)(a
bc
4
2
a
bx
2
cbxaxxf 2)(
a
bc
4
2
a
bx
2
dcxbxaxxf 23)(
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Here, all a’s are constants and .
This function is called the general polynomial of degree n with coefficients
.
Numerous problems in mathematics and its applications involve polynomials. Often one is
very interested in finding the number of zeros in P(x). This means, the values of x such
that P(x) = 0. This equation:
is called the general equation of degree n. This equation has at most n real solutions
or roots, but it doesn't need to have any.
According to the fundamental theorem of algebra, every polynomial of the form
can be written as a product of polynomials of degree
1 or 2. For example, we can re-write as .
Factoring Polynomials
Let P(x) and Q(x) be two polynomials for which . Then there always exist
unique polynomials q(x) and r(x) such that:
where the degree of r(x) is less than the degree of Q(x). This fact
is called the remainder theorem.
When x is such that . Then can be written in the form:
An important and special case is when Q(x)=x-a. Then Q(x) is of the degree 1, so the
remainder r(x) must have degree 0, and is therefore a constant. In this special case, for
all x:
In particular, when x=a , P(a)=r. Hence, x – a devides P (x) if and only if P(a) = 0.This
can be formulated as:
Polynomial division
You can devide polynomials in much the same way as you use long division to devide
numbers. It is possible that a devision gives a solution but leaves a remainder. For an
example consider the compulsory literature.
Rational functions
A rational function is a function: that can be expressed as the ratio
of two polynomials P(x) and Q(x). This function is defined for all x where . The
ratio is proper if the degree of P(x) is less than the degree of Q(x). When the degree of
P(x) is more than the degree of Q(x), it is called an improper rational function.
01
1
1 ...)( axaxaxaxP n
n
n
n
0na
naaa ,...,, 10
01
1
1 ...)( axaxaxaxP n
n
n
n
123 xxx )1)(1( 2 xx
)()( xQxP
)()()()( xrxQxqxP
0)( xQ )()()()( xrxQxqxP
)(
)()(
)(
)(
xQ
xrxq
xQ
xP
raxxqxP ))(()(
0)( aPax
)(/)()( xQxPxR
0)( xQ
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Power Functions
The general power function f is defined by the formula:
(Where x>0 and r and A are constants).
Graphs of Power Functions
The shape of the graph depends crucially on the
value of r. See the example below:
Exponential Functions
A quantity that increases/decreases by a fixed
factor per unit of time is said to
increase/decrease exponentially. If the factor
is a, this leads to the following exponential
function:
(A and a are positive constants)
Beware of the fundamental difference between:
en .
The first one is an exponential function where
the exponent varies. The second one is an
example of a power function where the base
varies and the exponent is a constant.
The doubling time is the time it takes for the result of an exponential function to
double. Suppose the function:
The value at t=0 is A. The doubling time t* is given by the equation or
after cancelling A; . Thus, the doubling time of the exponential function
is the power to which a must be raised in order to get 2.
The Natural Exponential Function
The most important base for an exponential function is denoted by the letter e
(=2,7182818284590…).
Given the base e the natural exponential function is:
Logarithmic Functions
Like we said before, the doubling time of an exponential function was
defined as the time it takes for f(t) to become twice as large:
To solve this, we use the natural logarithm. Therefore, we introduce the following
useful definition: If , we call u the natural logarithm of a, and we write u=ln a.
Hence, we have the following definition of the symbol ln a:
Here, a can be all positive values. When , then x is: ln 7.
rAxxf )(
tAatf )(
xaxf )(axxg )(
tAatf )(
AAatf t 2*)( *
2* tatAatf )(
xexf )(
**)( tAatf
2* ta
aeu
ae a ln
7xe
Overview Study Material Business Mathematics 1 2015-2016
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18
There are a couple of rules for the natural logarithmic function ln:
(x and y are positive);
(x and y are positive);
(x is positive);
;
There are no simple formulas for en .
The Function g(x)=ln(x)
For each positive number x, the number ln x is defined by: . So, the u =ln x is
the solution to the equation e^u=x. We call the resulting function the natural
logarithm of x:
(x>0)
The graph of the function can be found in the compulsory study material.
Logarithms with Bases other than e
Suppose a (a>1). If , then we call u the logarithm of x to base a, and we write:
where is also true.
By taking the ln on each side of , we obtain , so that:
Log a obeys the same rules as ln:
;
;
;
.
yxxy lnln)ln(
yxy
xlnln)ln(
)ln()ln( xpx p
xeexe xx ln,,1ln,01ln ln
)ln( yx )ln( yx
xe x ln
xxg ln)(
xau
xu alog xaxa
log
xaxa
log
xa
xa lnln
1log
yxxy aaa loglog)(log
yxy
xaaa loglog)(log
)(log)(log xpx a
p
a
1log,01log aaa
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