Transcript Lecture 3 - Personal Webpages (The University of Manchester)

TX-1037 Mathematical Techniques for Managers
Dr Huw Owens
Room B44 Sackville Street Building
Telephone Number 65891
Dr Huw Owens - University of Manchester – February 2006
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•
•
•
•
•
•
Functions of more than One Variable
Economic Variables and Functions
Total and Average Revenue
Total and Average Cost
Profit
Production Functions, Isoquants and the average
Product of Labour
• Equations in economics
• Rewriting and solving equations
• Substitution
Dr Huw Owens - University of Manchester – February 2006
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Functions of more than one variable
• Multivariate function: the dependent variable, y, is a
function of more than one independent variable.
• if y=f(x,z) y is a function of the two variables x and
z.
• We substitute values for x and z to find the value of
the function.
• If we hold one variable constant and investigate the
effect on y of changing the other, this is a form of
comparative statistical analysis.
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Some definitions (1)
• Total revenue, TR, is the amount of money received
by the firm from the sale of goods and total cost, TC,
is the amount of amount of money that the firm has
to spend to produce these goods.
• The profit function is denoted by the Greek letter ∏
(pronounced “pie”) and is defined to be the difference
between total revenue, TR, and total cost, TC.
• Total cost, TC, function relates to the production costs
to the level of output, Q.
• However, in the short run some of these costs are
fixed.
• Fixed costs, include the cost of land, equipment, rent
and possibly skilled labour.
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Some definitions (2)
• Obviously, in the long run all costs are variable but
these particular costs take time to vary, so can be
though of as fixed in the short run.
• Variable costs vary with output and include the cost of
raw materials, components, energy and unskilled
labour.
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Definitions – Factors of production
• The output Q, of any production process depends on a
variety of inputs, known as factors of production.
These include land, capital, labour and enterprise.
• For simplicity. We restrict our attention to capital and
labour.
• Capital, K, denotes all man-made aids to production
such as buildings, tools and plant machinery.
• Labour, L, denotes all paid work in the production
process
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Total and average revenue
• When a firm sells a quantity, Q, of goods each at a
price P, its total revenue, TR, is the price that is paid
multiplied by the quantity sold.
• TR = P*Q
• Average return, AR, is the revenue received by the
firm per unit of output sold. This is the total revenue
divided by the quantity sold.
• AR = TR/Q = P
• The average return curve shows the average revenue
or price at which different quantities are sold. It
shows the prices that people will pay to obtain various
quantities of output and so is known as the demand
curve.
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Total and average revenue
• A market demand curve is assumed to be downward
sloping.
• Different prices are associated with different
quantities being sold at lower prices.
• There will also be an associated downward sloping
marginal, MR, curve but we will investigate this in
later lectures.
• For example, If average revenue is given by P=72-3Q
sketch this function and also, on a separate graph, the
total revenue function.
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Linear average revenue function
• The average revenue function has P on the vertical axis and
Q on the horizontal axis. The general form of the linear
equation is y=ax+b. We only need to find two points in
order to sketch the function. P=72-3Q
80
0, 72
70
60
P
50
10, 42
40
P
30
20
20, 12
10
0
0
5
15
10
20
25
Q
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• Next we need to find an expression for TR.
• TR = P*Q =(72-3Q)Q = 72Q-3Q2
Q=72-3Q
500
450
12, 432
10, 420 14, 420
8, 384
16, 384
400
350
6, 324
TR
300
250
TR
4, 240
200
150
2, 132
100
50
0
-2
0, 0
24, 0
3
8
13
18
23
28
Q
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• The function is a quadratic, so we must find a number
of points.
• The graphs shows a curve which at first rises
relatively steeply, then flattens out and reaches a
maximum Q=12, after which it falls. The curve is
symmetric. Its shape to the right of its maximum
value is the mirror image of that to the left.
• Symmetric: the shape of one half of the curve is the
mirror image of the other half.
• Some firms sell their output at the same price. This is
a feature of firms operating under the market
structure known as perfect competition.
• These firms face a horizontal demand curve and have
a total revenue function which is an upward sloping
straight line passing through the origin.
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Total and Average Cost
• A firm’s total cost of production, TC, depends on its
output, Q.
• The TC function include may a constant term, which
represents fixed costs, FC.
• The part of the TC that varies with Q is called variable
cost, VC.
• FC is the constant term in TC
• VC = TC-FC
• Average Total Cost, AC = TC/Q
• Average Variable Cost, AVC = VC/Q
• Average Fixed Cost, AFC = FC/Q
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Example
• For a firm with total cost given by,
• TC = 120+45Q-Q2+0.4Q3
• Find AC, FC, VC, AVC and AFC functions. List some
values of TC, AC and AFC, correct to the nearest
integer. Sketch the total cost function and on a
separate graph, the AC and AFC functions.
• AC = TC/Q = 120/Q+45-Q+0.4Q2
• FC= 120 (the constant term in the TC)
• VC = TC-FC = 45Q-Q2+0.4Q3
• AVC = VC/Q = 45-Q+0.4Q2
• AFC = FC/Q = 120/Q
Dr Huw Owens - University of Manchester – February 2006
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Total Cost
Total Cost
2500
2000
1500
£
Total Cost
1000
500
0
0
5
10
15
20
Q
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• The average total cost curve as first falls as output
rises, but later the curve rises again.
• Average fixed cost is always declining as output
increases.
500
Costs (£)
400
300
AC
AFC
200
100
0
0
5
10
15
20
Q
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Profit
• Profit is the excess of a firm’s total revenue, TR, over its
total cost, TC, and so we calculate it by subtracting TC from
TR. Using the symbol ∏ as the variable for profit.
• ∏ = TR-TC
• A firm has the total cost function
• TC = 120+ 45Q-Q2+0.4Q3
• And faces a demand curve given by
• P=240-20Q
• What is its profit function?
• TR = PQ = 240Q-20Q2
• Since TC comprises several terms we enclose it in brackets
as we substitute
• =240Q-20Q2 – (120+45Q-Q2+0.4Q3)
• =-120+195Q-19Q2-0.4Q3
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Production functions, isoquants and the average
product of labour
• A production function shows the quantity of output
(Q) obtained from specific quantities of inputs,
assuming they are used efficiently. Q=f(L,K)
• In the short run the quantity of capital is fixed.
• In the long run both labour (L) and capital (K) are
variable.
• Plot Q on the vertical axis against L on the
horizontal for a short-run production function.
• Plot K against L and connect points that generate
equal output for an isoquant map.
• Average Product of Labour (APL) = Q/L
• An isoquant connects points at which the same
quantity of output is produced using different
combinations of inputs.
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An example
• A firm has the production function Q=25(LK)2-0.4(LK)3.
• If K=1, find the values of Q for L=2,3,4,6,12,14 and 16.
Sketch this short-run production function putting L and Q on
the axes of your graph.
Production functions for different values of K
15000
14464.8
13737.6
13363.2
10000
Q
K=1
K=2
K=3
5767.2
5000
2908.8
1733.4
813.6
0
0
5
10
15
20
L
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• Now sketch another representation of this production
function as an isoquant map. Plot L and K on the axes and
look for combinations of L and K amongst the values you
have calculated which give the same value of Q. Such
points lie on the same isoquant.
Isoquant map
6
5
Q=814
Q=2909
Power (Q=814)
Power (Q=2909)
K
4
3
2
1
0
0
5
10
15
L
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• For a short-run production function with K = 3, find and plot
the average product of labour function.
• For K=3 we have Q=25(3L)2-0.4(3L)3=225L2-10.8L3
• APL=Q/L = 225L-10.8L2
The average product of labour function
1200
APL
800
APL
400
0
0
5
10
15
20
L
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Equations in economics
• An equation is a statement that two expressions are
equal to one another.
• In economic modelling we express relationships as
equations and then use them to obtain analytical
results. Solving the equations gives us values for
which the equations are true.
• Solving equations lets us discover where curves
intersect. Economists are often interested in these
points because they may provide information about
equilibrium situations.
• Graphical solutions can be obtained by reading off the
x and y values at the point or points of intersection
BUT the results have limited accuracy.
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Rewriting and solving equations
• Since the x and y values are the same on both curves at
intersecting points we can obtain an exact solution using
algebra.
• At an intersection of the functional relationships y=f(x) and
y=h(x), the two y values are equal and therefore f(x)=h(x)
• Transposition: rearranging an equation so that it can be
solved, always keeping what is on the left of the equals sign
equal to what is on the right.
• For example, to solve for x the equation
• 140+6x = -30x+284
• 6x+140-140=-30x+284-140
• 6x=-30x+144
• 6x+30x=-30x+144+30x
• 36x=144
• x=144/36 = 4
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• When rewriting equations
• Add to or subtract from both sides.
• Multiply or divide through the whole of each side
(DON’T divide by zero).
• Square or take the square root of each side.
• Use as many stages as you wish.
• Take care to get all the signs correct.
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Example
• Plot the equations y=-5+2x and y=30-3x. At what values
of x and y do they cross? Find the algebraic solution by
setting the two expressions in x equal to one another.
• -5+2x = 30-3x
• 5x = 35, x=7
• When x=7, y = 9.
Lines intersect at (7,9)
30
25
20
y
15
y=-5+2x
y=30-3x
10
5
0
-5 0
2
4
6
8
10
12
-10
x
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Problem
8x2
• Solve for x, 4 x  16 
2x  3
•
•
•
•
•
(4x-16)(2x+3) = 8x2
8x2+12x-32x-48=8x2
-20x-48=0
-20x = 48
x = -2.4
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Solution in terms of other variables
• Not all equations will have numerical solutions. Sometimes
when you solve an equation for x you obtain an expression
containing other variables.
• If you are given a relationship of the form y=f(x), rewriting
the equation in the form x=g(y) is called finding the inverse
function. To be able to find the inverse function there must
be just one x value corresponding to each y value.
• For non-linear functions there can be difficulties in finding
an inverse but it may be done for a restricted set of values
(e.g. square roots).
• For the linear functions often used in economic models
inverse functions can always be found.
• One reason for finding the inverse function is if the variable
represented by y is conventionally plotted in economics on
the horizontal axis. Demand and supply equations are
examples of this.
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• Inverse function: expresses x as a function of y
instead of y as a function of x.
• Solve for x in terms of z, x=60+0.8x+7z
• 0.2x = 7z+60, x = 35z+300
• Solve y = x1/2+5, obtain an expression for x in terms
of y.
• X1/2=y-5, x = (y-5)(y-5), x = y2-10y+25
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Substitution
• Substitution: to write one expression in place of
another.
• When substituting, always be sure to substitute the
whole of the new expression and combine it with the
other terms in exactly the same way that the
expression it replaces was combined with them.
• For example, if y=x2+6u and x = 30-u, find an
expression for y in terms of u. Substituting 30-u for x
we obtain
• y=(30-u)2+6u, y=900-60u+u2+6u, y = u2-54u+900
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Demand and supply
• Demand and supply functions in economics express
the quantity demanded or supplied as a function of
price, Q=f(P).
• According to mathematical convention the dependent
variable, Q, should be plotted on the vertical axis.
• Economic analysis, however, uses the horizontal axis
as the Q axis and for consistency we will follow that
approach.
• So that we can determine the points on the graph in
the usual way, before plotting a demand or supply
function we first find its inverse function giving P as a
function of Q.
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Demand and supply
• Find the inverse function for the demand equation Q=80-2P
and sketch the demand curve.
• P=40-Q/2
Demand Curve P=40-Q/2
40
P
30
P=40-Q/2
20
10
0
0
20
40
60
80
100
Q
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Market equilibrium
• Market equilibrium occurs when the quantity supplied
equals the quantity demanded of a good. The supply and
demand curves cross at the equilibrium price and quantity.
• If you plot the demand and supply curves you can read off
the approximate equilibrium values from the graph.
• Another way is to solve algebraically for the point where
demand and supply are equal.
• For example, quantity demanded, Qd, is given by
• Qd=96-4P, and quantity supplied, Qs, is given by
• Qs = 8P
• In equilibrium, Qs = Qd so by substitution
• 8P = 96-4P
• 12P=96, P=8, the equilibrium price
• Substitute P into either the supply or demand equation.
Using the supply equation gives, Qs=96-4P, Qs=96-32,
Qs=64
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Changes in demand or supply
• The quantity demanded and the quantity supplied of a good
are usually modelled as dependent on a number of factors.
• All of these except price are assumed to be constant when
the demand and supply curves are drawn.
• Changes in factors other than price alter the position of the
curves.
• For example, suppose we have a multivariate demand
function where Q, the demand for good X depends on P the
price of X, together with M, consumer income and Pz, the
price of another good,Z. The demand function is
• Q1=80-5P+0.1M+0.3Pz when M and Pz are fixed
respectively at 2500 and 60 the demand curve becomes
• Q2=80-5P+0.1(2500)+0.3(60) = 298-5P
• We get a new demand curve, which is parallel to the old
one but shifted down from it.
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Changes in demand or supply
• Demand and supply curves can also change their
shapes in other ways. For example, suppose a
change in tastes causes twice as much to be
demanded at any price. If the original demand is
given by
• Q=55-5P
• Denoting the new quantity demanded Q2, we know
that it is twice Q and so,
• Q2=2Q = 2(55-5P) = 110-10P
• Again the use of subscripts distinguishes two different
curves.
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Changes in demand and supply - Example
• For the demand and supply functions given, find the
inverse functions giving P as a function of Q, sketch
the demand and supply curves and mark the
equilibrium position.
• Demand: Qd=110-5P
• Supply: Qs=6P
• If demand increases by 20%, find the new demand
function, its inverse and the new equilibrium position
on the diagram.
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Changes in demand or supply - Example
• Rewrite the demand equation, adding (5P-Qd) to both
sides. This gives
• 5P=110-Qd
• Dividing both sides by 5 gives the inverse demand
function
• P=22-Qd/5
• For the inverse of the supply equation we interchange
the sides, obtaining
• 6P=Qs
• And then divide by 6 to get
• P=Qs/6
• The demand and supply curves are shown on the
following figure.
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• Equilibrium occurs where they cross, and here the
values of P and Q on the demand curve equal those
on the supply curve.
• This is the point Q=60, P=10.
• When demand increases, the quantity demanded is
20% greater than before at every price. Hence the
new quantity demanded, Qd2, is given by
• Qd2 = 1.2(110-5P)=132-6P
• The inverse function is
• P=22-Qd2/6
• The new equilibrium position occurs where this curve
crosses the supply curve at the point Q=66, P=11
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Equilibrium P and Q increase when demand increases by
20%
30
25
20
P
15
P=Qs/6
P=22-Qd/5
P=22-Qd/6
10
5
0
-5
0
50
100
150
200
-10
Q
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Cost–Volume–Profit (CVP) Analysis
• Two simplifying assumptions are made: namely that
price and average variable costs are both fixed
 = P.Q – (FC + VC) = P.Q – FC – VC
• Multiplying both sides of the expression for AVC by Q
we obtain
AVC.Q = VC and substituting this
 = P.Q – FC – AVC.Q
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Special Assumptions of CVP Analysis
•
•
•
•
P is fixed
AVC is fixed
 is a function of Q but P, FC, and AVC are not
We can write the inverse function expressing Q as a
function of 
• Adding FC to both sides gives
 + FC = P.Q – AVC.Q
• Interchanging the sides we obtain
P.Q – AVC.Q =  + FC
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Solving for Desired Sales Level
•
•
•
•
•
Q is a factor of both terms on the left so we may write
Q(P – AVC) =  + FC
Dividing through by (P – AVC) gives
Q = ( + FC)/(P – AVC)
If the firm’s accountant can estimate FC, P and AVC,
substituting these together with the target level of
profit, , gives the desired sales level
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Linear Equations
• Slope of a line: distance up divided by
distance moved to the right between any two
points on the line
• Coefficient: a value that is multiplied by a
variable
• Intercept: the value at which a function cuts
the y axis
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Representing a Line as y = mx + b
•
•
•
•
•
•
The constant term, b, gives the y intercept
The slope of the line is m, the coefficient of x
Slope = y/x = (distance up)/(distance to right)
Lines with positive slope go up from left to right
Lines with negative slope go down from left to right
Parameter: a value that is constant for a specific function
but that changes to give other functions of the same type;
m and b are parameters
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A horizontal line has zero slope
as x increases,
y does not change
30
y
20
y = 18
slope = 0
10
0
0
5
Dr Huw Owens - University of Manchester – February 2006
x 10
44
Positive slope, zero intercept
y = 9x
500
as x increases,
y increases
y
250
slope = 9
line passes through the origin
0
0
25
Dr Huw Owens - University of Manchester – February 2006
x
50
45
Negative slope, positive intercept
y
60
50
larger x values go with
smaller y values
40
30
slope = - 4
20
y = 50 - 4x
10
0
0
5
10
Dr Huw Owens - University of Manchester – February 2006
x 15
46
Positive slope, negative intercept
y
40
30
20
10
as x increases,
y increases
0
-10
-20
-30
y = -25 + 3x
slope = 3
x
10
20
line cuts y axis below the origin
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A vertical line has infinite slope
40
x = 15
y
y increases but x does not change
30
20
10
slope = 
0
0
5
10
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15
x
20
48
Budget Line
• If two goods x and y are bought
the budget line equation is x.Px + y.Py = M
• To plot the line, rewrite as
y = M/Py – (Px/Py )x
• Slope = – Px/Py
the negative of the ratio of the prices of the goods
• Intercept = M/Py
the constant term in the equation
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The Parameters of a Budget Line
• Changing Px rotates the line about the point where it cuts
the y axis
• If Py alters, both the slope and the y intercept change
• the line rotates about the point where it cuts the x axis
• An increase or decrease in income M alters the intercept but
does not change the slope
• the line shifts outwards or inwards
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Constant Substitution Along a Line
• The rate at which y is substituted by x is constant
along a downward sloping line, but not along a curve
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Diminishing Marginal Rate of Substitution Along an
Indifference Curve
• Indifference curve: connects points representing different
combinations of two goods that generate equal levels of
utility for the consumer
• Diminishing marginal rate of substitution: as a consumer
acquires more of good x in exchange for good y, the rate
at which he substitutes x for y diminishes because he
becomes less willing to give up y for a small additional
amount of x
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Quadratic Equations
• A quadratic equation takes the form
ax2 + bx + c = 0
• You can solve it graphically
• or sometimes by factorizing it
• or by using the formula
where a is the coefficient of x2, b is the coefficient of x and
c is the constant term
 b  b2  4ac
x
2a
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Intersection of MC with MR or AVC
• Quadratic equations arise in economics where a quadratic
function, say marginal cost, cuts another quadratic
function, say average variable cost, or cuts a linear
function, say marginal revenue
• Equate the two functional expressions
• Subtract the right-hand side from both sides so that the
value on the right becomes zero
• Collect terms
• Solve the quadratic equation
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Simultaneous Equations
• Simultaneous equations can usually (but not always)
be solved if
number of equations = number of unknowns
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Solving Simultaneous Equations
• Solution methods for two simultaneous equations
include
• Finding where functions cross on a graph
• Eliminating a variable by substitution
• Eliminating a variable by subtracting
(or adding) equations
• Once you know the value of one variable, substitute it
in the other equation
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Simultaneous Equilibrium in Related Markets
• Demand in each market depends both on the price of the
good itself and on the price of the related good
• To solve the model use the equilibrium condition for each
market
demand = supply
• This gives two equations (one from each market) in two
unknowns which we then solve
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Exponential Functions
• Exponential function: has the form ax where the base, a, is
a positive constant and is not equal to 1
• The exponential function most used in economics is
y = ex
• The independent variable is in the power and the base is
the mathematical constant
e = 2.71828…
• Use your calculator or computer to evaluate ex
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Logarithmic Functions
• Logarithm: the power to which you must raise the base to
obtain the number whose logarithm it is
• Common logarithms denoted log or log10
are to base 10
• Natural logarithms denoted ln or loge are to base e and are
more useful in analytical work
• Equal differences between logarithms correspond to equal
proportional changes in the original variables
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Working with Logarithms
•
•
•
•
•
log (xy) = log (x) + log (y)
log (x/y) = log (x) – log (y)
log (xn) = n log (x)
ln (ex) = x
The reverse process to taking the natural logarithm is
to exponentiate
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Solving a Quadratic Equation in Excel
• You can enter formulae in Excel to calculate the two
possible solutions to a quadratic equation
• First calculate the discriminant b2 – 4ac
• To make your formulae easier to understand, name cells a,
b, const and discrim and use the names instead of cell
references
• By default, names are interpreted in Excel as absolute cell
references
• To name a cell, select it, type its name in the Name Box
and press the Enter key
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Plotting and Solving Equations in Excel
• Excel includes many inbuilt functions that you can type in
or access by clicking the Paste Function button
• Those for exponentials and logarithms are
=EXP()
and
=LN()
where the cell reference for the value to which the function
is to be applied goes inside the brackets
Dr Huw Owens - University of Manchester – February 2006
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Solving Equations with Excel Solver
• Excel includes a Solver tool designed to solve a set of
equations or inequalities
• Set out the data in a suitable format
• Interact with the Solver dialogue box to find the solution
• Excel does not solve the equations the same way as you
do when working by hand
• It uses an iterative method, trying out different possible
values for the variables to see if they fit the specified
requirements
Dr Huw Owens - University of Manchester – February 2006
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