Notes (Module: Science Skills and Safety)
IJSO Training: Science Skills and Safety
Notes
1.
Significant Digits
Suppose you want to find the volume of a lead cube. You could measure the length l of
the side of a lead cube to be 1.76 cm and the volume l 3 from your calculator reads 5.451776.
The measurement 1.76 cm was to three significant figures, so the answer can only be three
significant figures. So that the volume = 5.45 cm3. The following rules are applied
generally:

All non-zero digits are significant. (e.g., 22.2 has 3 sf)

All zeros between two non-zero digits are significant. (e.g., 1007 has 4 sf)

For numbers less than one, zeros directly after the decimal point are not significant. (e.g.,
0.0024 has 2 sf)

A zero to the right of a decimal and following a non-zero digit is significant. (e.g.,
0.0500 has 3 sf)

All other zeros are not significant. (e.g., 500 has 1 sf, 17000 has 2 sf)

When multiplying and dividing a series of measurements, the number of significant
figures in the answer should be equal to the least number of significant figures in any
data of the series.
For example, if you multiply 3.22 cm by 12.34 cm by 1.8 cm to find the volume of a piece of
wood, you get an initial answer 71.52264 cm3 from your calculator. However, the least
significant measurement is 1.8 cm with 2 sf. Therefore, the correct answer is only 72 cm3.

When adding and subtracting a series of measurements, the answer has decimal places
with the least accurate place value in the series of measurements.
For example, what is your answer by adding 24.2 g and 0.51 g and 7.134 g? You get an initial
answer 31.844 g from your calculator. However, the least accurate place measurement is
24.2g with only one decimal point. So the answer is 31.8 g.
Notes (Module: Science Skills and Safety)
Exercises:
1.1
How many significant figures are indicated by each of the following?
(a). 1247
(b). 1007
(c). 0.034
(d). 1.20  107
(e). 62.0
(f). 0.0025
(g). 0.00250
(h). sin 45.2o
(i). tan-10.24
(j). 3.2  10 16
1.2
(a) Add the following lengths of 2.35 cm and 7.62 m and 14.2 m.
(b) A rectangular box has lengths of 1.52 cm, 103.4 cm and 3.1 cm. Calculate the
volume of the box cm3.
(a) 0.0235 + 7.62 + 14.2 = 21.8435. The answer is 21.8 m
(b) 1.52 103.4  3.1  487.2208 . The answer is 490 cm3.
2.
Making Measurements
A measurement should always be regarded as an estimate. The precision of the final
result of an experiment cannot be better than the precision of the measurements made during
the experiment, so the aim of the experimenter should be to make the estimates as good as
possible.
There are many factors which contribute to the accuracy of a measurement. Perhaps the
most obvious of these is the level of attention paid by the person making the measurements: a
careless experimenter gets bad results! However, if the experiment is well designed, one
careless measurement will usually be obvious and can therefore be ignored in the final
analysis.
Systematic Errors
If a voltmeter is not connected to anything else it should, of course, read zero. If it does
not, the "zero error" is said to be a systematic error: all the readings of this meter are too high
or too low. The same problem can occur with stop-watches, thermometers etc. Even if the
instrument can not easily be reset to zero, we can usually take the zero error into account by
simply adding it to or subtracting it from all the readings. (However, other types of systematic
error might be less easy to deal with.)
For this reason, note that a precise reading is not necessarily an accurate reading. A
precise reading taken from an instrument with a systematic error will give an inaccurate
result.
Notes (Module: Science Skills and Safety)
Random Errors
Try asking 10 people to read the level of liquid in the same measuring cylinder. There
will almost certainly be small differences in their estimates of the level. Connect a voltmeter
into a circuit, take a reading, disconnect the meter, reconnect it and measure the same voltage
again. There might be a slight difference between the readings. These are random
(unpredictable) errors. Random errors can never be eliminated completely but we can usually
be sure that the correct reading lies within certain limits.
To indicate this to the reader of the experiment report, the results of measurements should
be written as
Result ± Uncertainty
For example, suppose we measure a length, l to be 25 cm with an uncertainty of 0.1 cm. We
write the result as
l  25.0  0.1 cm
By this, we mean that all we are sure about is that l is somewhere in the range 24.9 cm to
25.1 cm.
A.
Quantifying the Uncertainty
The number we write as the uncertainty tells the reader about the instrument used to make
the measurement. (As stated above, we assume that the instrument has been used correctly.)
Consider the following examples.
Example 1: Using a ruler
The length of the object being measured is obviously somewhere near 4.3 cm (but it is
certainly not exactly 4.35 cm). The result could therefore be stated as:
4.3 cm ± Half the smallest division on the ruler
In choosing an uncertainty equal to half the smallest division on the ruler, we are
accepting a range of possible results equal to the size of the smallest division on the ruler.
However, do you notice something which has not been taken into account? A measurement of
length is, in fact, a measure of two positions and then a subtraction. Was the end of the object
Notes (Module: Science Skills and Safety)
exactly opposite the zero of the ruler? This becomes more obvious if we consider the
measurement again, as shown below.
Notice that the left-hand end of the object is not exactly opposite the 2 cm mark of the ruler. It
is nearer to 2 cm than to 2.05 cm, but this measurement is subject to the same level of
uncertainty.
Therefore the length of the object is
(6.30 ± 0.05)cm - (2.00 ± 0.05)cm
so, the length can be between
(6.30 + 0.05) - (2.00 - 0.05) and (6.30 - 0.05) - (2.00 + 0.05)
that is, between
4.40 cm and 4.20 cm
We now see that the range of possible results is 0.2 cm, so we write
Length = 4.30 cm ± 0.10 cm
In general, we state a result as
Reading ± The smallest division on the measuring instrument
One may record the length of the following red stick to be 5.9  0.1 cm.
Notes (Module: Science Skills and Safety)
Example 2: Using a Stop-Watch
Consider using a stop-watch which measures to 1/100 of a second to find the time for a
pendulum to oscillate once. Suppose that this time is about 1s. Then, the smallest division on
the watch is only about 1% of the time being measured. We could write the result as
T = 1.00 ± 0.01s
which is equivalent to saying that the time T is between
0.99s and 1.01s
This sounds quite good until you remember that the reaction-time of the person using the
watch might be about 0.1s. Let us be pessimistic and say that the person's reaction-time is
0.15s. Now considering the measurement again, with a possible 0.15s at the starting and
stopping time of the watch, we should now state the result as
T = 1.00 s ± (0.01+ 0.3) s
In other words, T is between about 0.7s and 1.3s. We could probably have guessed the
answer to this degree of precision even without a stop-watch!
Conclusions from the preceding discussion
If we accept that an uncertainty (sometimes called an indeterminacy) of about 1% of the
measurement being made is reasonable, then
(a)
a ruler, marked in mm, is useful for making measurements of distances of about
100mm (or 10 cm) or greater.
(b)
a manually operated stop-watch is useful for measuring times of about 30 s or more
(for precise measurements of shorter times, an electronically operated watch should be
used)
Notes (Module: Science Skills and Safety)
B.
How many Decimal Places?
Suppose you have a timer which measures to a precision of 0.01s and it gives a reading of
4.58 s. The actual time being measured could have been 4.576 s or 4.585 s etc. However, we
have no way of knowing this, so we can only write
t = 4.58s ± 0.01s
We now repeat the experiment using a better timer which measures to a precision of
0·0001 s. The timer might still give us a time of 4.58s but now we would indicate the greater
precision of the instrument being used by stating the result as
t = 4.5800 s ± 0.0001 s
So, as a general rule, look at the precision of the instrument being used and state the result to
that number of decimal places.
C.
How does an uncertainty in a measurement affect the FINAL result?
The measurements we make during an experiment are usually not the final result; they are
used to calculate the final result. When considering how an uncertainty in a measurement
will affect the final result, it will be helpful to express uncertainties in a slightly different way.
Remember, the uncertainty in a given measurement should be much smaller than the
measurement itself. For example, if you write, "I measured the time to a precision of 0·01 s",
it sounds good: unless you then inform your reader that the time measured was 0·02 s ! The
uncertainty is 50% of the measured time so, in reality, the measurement is useless. We will
define the quantity Relative Uncertainty (sometimes called fractional uncertainty) as follows
(To emphasize the difference, we use the term "absolute uncertainty" where previously we
simply said "uncertainty").
We will now see how to answer the question in the title.
It is always possible, in simple
situations, to find the effect on the final result by straightforward calculations but the
following rules can help to reduce the number of calculations needed in more complicated
situations.
Rule 1:
If a measured quantity is multiplied or divided by a constant, then the relative
uncertainty stays the same. See Example 1.
Rule 2:
If two measured quantities are added or subtracted then their absolute
Notes (Module: Science Skills and Safety)
uncertainties are added. See Example 2.
Rule 3:
If two (or more) measured quantities are multiplied or divided then their relative
uncertainties are added. See Example 3.
Rule 4:
If a measured quantity is raised to a power then the relative uncertainty is
multiplied by that power. (If you think about this rule, you will realise that it is just
a special case of rule 3.) See Example 4.
A few simple examples might help to illustrate the use of these rules. (Rule 2 has, in fact,
already been used in the section "Using a Ruler" on page 3.)
Example 1
Suppose that you want to find the average thickness of a page of a book. We might find
that 100 pages of the book have a total thickness of T = 9.0 mm. If this measurement is made
using an instrument having a precision of 0·1 mm, then the relative uncertainty is e  0.1/ 9.0 .
Hence, the average thickness of one page, t, is given by t  T /100  0.09 mm with an
absolute uncertainty 0.09  e  0.001mm, or t = 0.090 mm ± 0.001mm.
Note: both T and t
have only 2 sf.
Example 2
(a) To find a change in temperature T  T2  T1 , in which the initial temperature T1 is found
to be 20°C ± 1°C and the final temperature T2 is found to be 45°C ± 1°C. Then
T  25  2 °C.
(b) Now, the initial temperature T1 is found to be 20.2°C ± 0.1°C and the final temperature T2
is found to be 45.23°C ± 0.01°C. Then the calculated value is 25.03  0.11 . However,
the least decimal place measurement is 20.2 with only one decimal point. So the answer
is T  25.0  0.1 °C.
Exercise:
2.1
The first part of the trip took 25  3 (s), the second part 17  2 (s). How long time did
the whole trip take? How much longer did the first part take compared to the second
part?
Notes (Module: Science Skills and Safety)
Example 3
To measure a surface area, S, we measure two dimensions, say, x and y, and then use
S  xy .
Using a ruler marked in mm, we measure x = 54 ± 1 mm and y = 83 ± 1 mm
This means the relative uncertainties of x and y are, respectively, 1/ 54 and 1/ 83 . The
relative uncertainty of S is then e  1/ 54 1/ 83  0.03 . The calculated value of the surface
area is 4482 with uncertainty 134.46. Thus, the surface area S is 4500 ± 100mm². (2
significant figures)
Exercises:
2.2
An object covers 602.0  1.0 (m) in 32.0  1.0 (s). What was the speed?
2.3
If using the formula v = u + at we insert u = 5.0  0.5 ms-1, a = 0.100  0.005 ms-2
and time t = 3.00  0.15 s, what will v be?
Example 4
To find the volume of a sphere, we first find its radius, r (usually by measuring its
4
diameter). We then use the formula: V   r 3 . Suppose that the diameter of a sphere is
3
measured (using an instrument having a precision of ±0.2mm) and found to be 50.0mm, so
r  25.0 mm with relative uncertainty 0.2/ 50  0.004 , so r = 25.0 ± 0.1mm. The relative
uncertainty of V is 3 0.004  0.012 . The volume of the sphere is V = 65500 ± 800 mm3. (3
significant figures)
Exercises:
2.4
If we use the formula x  y / z 2 and the percentage uncertainty (relative uncertainty
 100%) in y is 5% and in z 3%, what is it percentage uncertainty in x?
2.5
Same as above, but the formula is x  y / z ?
2.6
The dimensions of piece of paper are measured using a ruler marked in mm. The
results were x = 50 mm and y = 40 mm.
(a) Rewrite the results of these measurements "correctly".
(b)
Calculate the maximum and minimum values of the area of the sheet of paper
which these measurements give.
(c)
Express the result of the calculation of area of the sheet of paper in the form:
area = A ±
Notes (Module: Science Skills and Safety)
2.7
A body is observed to move a distance s = 15 m in a time t = 6 s. The distance was
measured using a ruler marked in cm and the time was measured using a watch giving
readings to 0.1 s
a)
Express these results "correctly" (that is, giving the right number of significant
figures and the appropriate indeterminacy).
b)
2.8
Use the measurements to calculate the speed of the body, including the
uncertainty in the value of the speed.
A body which is initially at rest, starts to move with acceleration a. It moves a distance
s = 10.00 ± 0.01 m in a time t = 4.0 ± 0.1 s. Calculate the acceleration.
2.9 The diameter of a cylindrical piece of metal is measured to a precision of ± 0.01 mm.
The diameter is measured at five different points along the length of the cylinder. The
results are shown below. Units are mm.
(i) 6, (ii) 6.4, (iii) 5.7, (iv) 6 and (v) 6.4
3.
(a)
Rewrite the list of results "correctly".
(b)
Calculate the average value of the diameter.
(c)
State the average value of the diameter in a way which gives an indication of
the precision of the manufacturing process used to make the cylinder.
(d)
Calculate the average value of the area of cross section of the cylinder. State
the result as area = A
mm2) .
Graphs
The results of an experiment are often used to plot a graph. A graph can be used to verify
the relation between two variables and, at the same time, give an immediate impression of the
precision of the results. When we plot a graph, the independent variable is plotted on the
horizontal axis. (The independent variable is the cause and the dependent variable is the
effect.)
A.
Straight Line Graphs
If one variable is directly proportional to another variable, then a graph of these two
variables will be a straight line passing through the origin of the axes. So, for example, Ohm's
Law has been verified if a graph of voltage against current (for a metal conductor at constant
temperature) is a straight line passing through (0,0). Similarly, when current flows through a
given resistor, the power dissipated is directly proportional to the current squared. If we
wanted to verify this fact we could plot a graph of power (vertical) against current squared
(horizontal). This graph should also be a straight line passing through (0,0).
Notes (Module: Science Skills and Safety)
B.
The "Best-Fit" Line
The "best-fit" line is the straight line which passes as near to as many of the points as
possible. By drawing such a line, we are attempting to minimize the effects of random errors
in the measurements.
For example, if our points look like this
The best-fit line should then be:
Notice that the best-fit line does not necessarily pass through any of the points plotted.
Notes (Module: Science Skills and Safety)
C.
To Measure the Slope of a Graph
The slope of a graph tells us how a change in one variable affects the value of another
variable. The slope of the graph is defined as
and must, of course, be stated in the appropriate UNITS.
(x1, y1) and (x2, y2) can be the co-ordinates of any two points on the line but for best precision,
they should be as far apart as possible as shown in the two examples below.
In the second graph, it is clear that y decreases as x increases so in this case, the slope is
negative.
Notes (Module: Science Skills and Safety)
D.
Error Bars
Instead of plotting points on a graph we sometimes plot lines representing the uncertainty
in the measurements. These lines are called error bars and if we plot both vertical and
horizontal bars we have what might be called "error rectangles", as shown below
x was measured to ±0·5s, y was measured to ±0·3m
The best-fit line could be any line which passes through all of the rectangles. Assume that
the line passes through zero, use the example above to estimate the maximum and minimum
slopes of lines which are consistent with these data. (The diagram is too small to expect
accurate answers but you should find about 1.06ms-1 maximum and about 0.92ms-1
minimum.)
Notes (Module: Science Skills and Safety)
E.
Measuring the slope at a Point on a Curved Graph
Usually we will plot results which we expect to give us a straight line.
If we plot a
graph which we expect to give us a smooth curve, we might want to find the slope of the
curve at a given point; for example, the slope of a displacement against time graph tells us the
(instantaneous) velocity of the object.
To find the slope at a given point, draw a tangent to the curve at that point and then find
the slope of the tangent in the usual way. This method is illustrated on the graph on the next
page. A tangent to the curve has been drawn at x = 3s. The slope of the graph at this point is
given by y / x = (approximately) 6ms-1.
Exercises:
3.1
The diameter of a metal ball is measured to be 30.0 mm ± 0.1 mm. The mass of the ball
is measured to be 110 g ± 1 g. Use these results to find a value for the density ρ, of the
metal of which the ball is made.
Density is defined as mass per unit volume so to calculate the density of a substance
we use the equation:
Notes (Module: Science Skills and Safety)
3.2
An investigation was undertaken to determine the relationship between the length of a
pendulum l and the time taken for the pendulum to oscillate twenty times. The time it
takes to complete one swing back and forth is called the period T. It can be shown that
l
T  2
where g is the acceleration due to gravity. The following data was
g
obtained.
Length of pendulum
 0.05 m
Time for 20 oscillations
 0.2 s
0.21
18.1
0.40
25.5
0.62
31.5
0.80
36.8
1.00
40.4
(a)
(b)
(c)
(d)
(e)
(f)
(g)
3.3
Period T
T2
Absolute error of
T2
Complete the period column for the measurements. Be sure to give the uncertainty and
the units of T.
Calculate the various values for T 2 including its units.
Determine the absolute error of T 2 for each value.
Draw a graph of T 2 against l. Make sure that you choose an appropriate scale to use
as much of a piece of graph paper as possible. Label the axes, put a heading on the
graph, and use error bars. Draw the curve of best fit.
What is the relationship that exists between T 2 and l ?
Are there any outliers?
From the graph determine a value of g.
Suppose the relationship between the semi-major axis a and the sidereal period P of a
planet in the solar system is given by a  kP , where k and α are constants. From the
table below, plot a suitable graph to find k and α.
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Semi-major
0.3871
0.7233
1.0000
1.5237
5.2028
9.5388
19.1914
30.0611
0.2408
0.6152
1.0000
1.8809
11.862
29.458
84.01
164.79
Axis a (AU)
Sidereal
Period P (year)
Notes (Module: Science Skills and Safety)
4. Writing Experimental Reports
You will perform a number of experiments in the future. You must keep a record of ALL
the experiments which you perform. For a few of the experiments you will be required to
present a full, detailed report, which will be graded. The grades will form part of your final
result (remember, a significant percentage of your final result will be based on your practical
abilities).
Title
1.
Usually, a report is set out as follows.
Introduction
Diagram
Method
Results
Theory
Conclusion
Title (Aim): The title must state clearly the aim of the experiment. It must tell the
reader what you are trying to prove or measure. For example, “Ohm’s Law” is not a
suitable title for an experiment report, whereas, “Experiment to verify Ohm’s Law” is
a suitable title. Similarly, “Relative Density” is not a suitable title but “Experiment to
measure the Relative Density of some common substances” is a more suitable title.
2.
Introduction: If the experiment is designed to verify a law, state the law in the
introduction. The introduction can also include such ideas as why the
results/conclusions of the experiment are important in every-day life, in industry etc.
(It might even include a little historical background, but not too much).
3.
Hypothesis: Before starting your investigation you usually have some idea of what
you expect the results will show. The hypothesis is basically a statement of what you
are expecting.
If you are trying, for example, to show how two variables are related,
state the expected relation and try to give and explanation of you choice. For
example, when Newton was thinking about gravitation, he assumed that the strength
of the force of gravity would become weaker as one moved further from the body
causing the field. He suggested that the relation between force and distance might
involve the inverse of the distance squared and to defend this choice he pointed out
that the surface area of a sphere depends on the square of its radius. Various
observations then confirmed this suggestion.
4.
Diagram: In most cases a labelled diagram is useful. Every electrical experiment
report must include a circuit diagram. If diagrams are drawn by hand, use a sharp
pencil and a ruler. (If you use a computer, learn how to make the best use of your
drawing program.)
Notes (Module: Science Skills and Safety)
5.
Method: The method section should give enough detail to enable another
experimenter to repeat the experiment to see if he/she agrees with your
results/conclusions. The method should include
6.

a description of the apparatus used

what measurements you made (if possible, in the order you made them)

what precautions you took to ensure the best accuracy possible

a mention of any unexpected difficulties (and how you overcame them)
Results: You should record all the measurements made during the experiment along
with some indication of the uncertainty of each measurement. Whenever possible,
present the results in the form of a table.
7.
Theory: This section should include any information which might help the reader to
understand how you used your measurements to reach the aim mentioned in the title.
For example, in one experiment, designed to measure the relative density of a
substance, the actual measurements made are two distances. The theory section of a
report on such an experiment must include a clear explanation of how these two
distances are related to the relative density of the substance being measured.
8.
Conclusion/Evaluation: Every experiment report must have a conclusion. If your
aim was to verify a law, state whether you have verified the law or not. If the aim was
to measure a particular quantity (e.g. relative density), give the final measured value
of the quantity in the conclusion.
In the case of an experiment designed to measure some well known physical constant
you should attempt to explain any difference between your result and the accepted
result. For example, if you find g = 9.5 ms-2, you should try to think of the most likely
cause of this obvious error.
If the experiment results were in some way unsatisfactory, try to suggest how the
investigation might be improved in order to improve accuracy of measurements or
range of data obtained. This evaluation section should include comments on the
apparatus used and the method employed.
Notes (Module: Science Skills and Safety)
5. General laboratory Rules
Do not enter the laboratory unless the laboratory superintendent is present.
When you come into the laboratory, you should walk to your place calmly.
If you run
you are sure to bump into someone - and if he is carrying equipment there could be an
accident.
At your place, take out your writing materials, file and text book.
Put your bag under
the bench and put your coat out of the way in a clean area. Do not leave anything lying beside
your bench; someone is sure to fall over it.
Wear a white laboratory jacket. This is important as the laboratory jacket will stop your
clothes from getting dirty or burnt.
If you have long hair make sure that you have an elastic band or a hair clip to tie it back.
This will help to keep it out of the way which is much safer.
Wear safety spectacles to protect your eyes when you are using chemicals such as acids
and when you are boiling liquids or heating solids.
On your bench you find a water tap and a sink.
The laboratory is not as clean as a
cafeteria, so do not drink from the taps in the laboratory. Also do not eat in the laboratory.
After you have been doing practical work, especially if you have been handling animals or
chemicals, wash your hands carefully.
You will also find a gas tap on your bench. This is for use with the Bunsen burner.
Sometimes you will need to heat things and that is what the Bunsen burner is used for. Your
teacher will show you how to use one properly. When you are not using a Bunsen burner the
gas tap should be turned off all the time.
The third thing that you will find on your bench is an electricity socket.
If you use the
electricity on your bench, for example when you use a lamp, make sure that the bench is dry.
Some practical investigations are wet, so drops of water can spread all over the bench and
your papers. It is a good idea to remove everything that you do not need from your bench
during practical work. You will only need a pencil or a pen to write with, a piece of paper for
your results and the instructions. The instructions can be kept safe in a plastic folder.
When you do a practical investigation you will need to collect equipment and materials.
Never carry too much equipment each time.
Notes (Module: Science Skills and Safety)
When you have finished a practical investigation always leave your bench clean and dry.
You can rinse and clean your test tubes in the sink but do not put solid objects down the sink,
they will block it.
When you leave the laboratory take all your belongings with you and make sure that
everything is turned off (gas tap, water tap).
If you should have an accident always tell the laboratory superintendent.
Exercises:
Filling in the missing words.
1. Put all coats in a ....................................
2. Your bags should be put under ....................................
3. Never .................................... or shout in a laboratory.
4. Do not use the services (gas, water or ....................................) unless you are told to.
5. Never .................................... or drink in a laboratory
6.
.................................. long hair during practical work.
7. Protect your eyes with .................................... when you boil liquids
8. Put all solid waste in the .................................... not in the ....................................
9. Put broken glass in the ....................................
10. Report all ................................... immediately to the laboratory superintendent.
11. Wash your hands after handling .................................... and ....................................
clean and out of the way area, the bench, run, electricity, eat, tie back, safety spectacles, plastic waste bin,
sink, metal waste bin, accidents, chemicals, animals.
Notes (Module: Science Skills and Safety)
Supplementary Notes
Indices and Logarithm
Scientists investigate the dependence of two or more physical or chemical or biological
quantities. For example, the relationship between a current I flowing through a light bulb with
resistance R and the potential difference V across the light bulb is simply V  IR . However,
physical situations are usually not simple, so mathematical tools, like indices and logarithms,
may be useful.
Laws for Indices
If a, b are real numbers and m, n are positive integers, we have the following laws:
(a)
a m  a n  a mn
(b)
am
 a mn
n
a
(c)
a 
(d)
a m  b m   ab 
(e)
am
a

 
bm
b
(f)
a 0  1 , where a  0
(g)
am 
(h)
m n
 a mn
m
m
1
, where a  0
am
a2  a
Notes (Module: Science Skills and Safety)
Properties of Logarithm
Definition: If a x  N , then x  log a N .
(a)
log a 1  0
(b)
log a a  1
(c)
loga  MN   loga M  log a N
(d)
M
log a 
N
(e)
log a M p  p log a M
(f)
log a N 

  log a M  log a N

logb N
, where a  1 , N  0
logb a
In physics, engineering and economics, the natural logarithms are most often used.
Natural logarithms use the base e  2.71828... , so that given a number e x , its natural
logarithm is x . For example, e3.6888 is approximately equal to 40, so that the natural
logarithm of 40 is about 3.6888 . The usual notation for the natural logarithm of x is ln( x)
and for logarithms to the base 10 is log( x) .
Example 1.1: It is known that Y  kX n . From the graph given, find k and n.
n = -3/7 and k = 1000
1
Example 1.2: Solve ln( x 2  3 x  2)  2 ln  2 x  1  ln 4
2
x = 5/7