Embed Size (px)
Citation preview
Revision
Logarithms & ExponentialsHigher Mathematics
www.maths4scotland.co.uk
Next
Logarithms Revision
Back NextQuit
Reminder
All the questions on this topic will depend
upon you knowing and being able to use,
some very basic rules and facts.
Click to show
When you see this buttonclick for more information
Logarithms Revision
Back NextQuit
Three Rules of logs
log log loga a ax y xy
log log loga a a
xx y
y
log logpa ax p x
Logarithms Revision
Back NextQuit
Two special logarithms
log 1a a
log 1 0a
Logarithms Revision
Back NextQuit
Relationship between log and exponential
log ya x y a x
Logarithms Revision
Back NextQuit
Graph of the exponential function
Logarithms Revision
Back NextQuit
Graph of the logarithmic function
Logarithms Revision
Back NextQuit
Related functions of
( )
( )
( )
( )
( )
( )
y f x a
y f x a
y f x
y f x
y f x a
y f x a
Move graph left a units
Move graph right a units
Reflect in x axis
Reflect in y axis
Move graph up a units
Move graph down a units
( )y f x
Click to show
Logarithms Revision
Back NextQuit
Calculator keys
ln = loge
log = 10log
Logarithms Revision
Back NextQuit
Calculator keys
ln=log 2.5e 2 . 5 = = 0.916…
log=10log 7.6 7 . 6 = = 0.8808…
Click to show
Logarithms Revision
Back NextQuit
Solving exponential equations
2.4 3.1 xe
Show
log 2.4 log 3.1 xe e eTake loge both sides
log 2.4 log 3.1 log xe e e e
log 2.4 log 3.1 loge e ex e Use log ab = log a + log b
log 2.4 log 3.1e ex
Use log ax = x log a
Use loga a = 1 log 2.4 log 3.1e e x
0.25593... (2dp)0.26
Logarithms Revision
Back NextQuit
60 80 keSolving exponential equations
log 60 log 80 ke e eTake loge both sides
log 60 log 80 log ke e e e
log 60 log 80 loge e ek e
Use log ab = log a + log b
log 80 log 60e ek
Use log ax = x log a
Use loga a = 1 log 60 log 80e e k
0.2876... (2dp)0.29
Show
Logarithms Revision
Back NextQuit
Solving logarithmic equations
3log 0.5y 0.53y Change to exponential form
Change to exponential form
1
21
2
1 13
33
y
(2dp)0.577.... 0.58y
Show
Logarithms Revision
Back NextQuit
3log (2 ) 2 log (3 )e ee elog loge eA B C
Simplify
expressing your answer in the form
where A, B and C are whole numbers.
3 2log (2 ) log (3 )e ee e 3 2log 8 log 9e ee e
3
2
8log
9ee
e
8log
9e
e
log 8 log log 9e e ee 1 log 8 log 9e e
Show
Logarithms Revision
Back NextQuit
5 5 5log 2 log 50 log 4 Simplify
5
2 50log
4
5log 25
25log 5 52 log 5 2
Show
Logarithms Revision
Back NextQuit
Find x if 4 log 6 2 log 4 1x x
4 2log 6 log 4 1x x 4
2
6log 1
4x
36log x
936
9
41
41 1 log 81 1x
1 81x 81x
Show
Logarithms Revision
Back NextQuit
5 5log 3 log 4x Given find algebraically the value of x.
5log 3 4x 5log 12x
5 12x
10 10log 5 log 12x 10 10log 5 log 12x
10
10
log 12
log 5x 1.5439..x (2dp)1.54x
Show
Logarithms Revision
Back NextQuit
Find the x co-ordinate of the point where the graph of the curve
with equation 3log ( 2) 1y x intersects the x-axis.
When y = 0 30 log ( 2) 1x
31 log ( 2)x
Exponential form
Re-arrange
13 2x
12 3x Re-arrange1
32x
Show
Logarithms Revision
Back NextQuit
The graph illustrates the law
ny kxIf the straight line passes through A(0.5, 0)and B(0, 1).
Find the values of k and n.
5 5log log ny kx
5 5 5log log logy k n x
5logY k nX
Gradient1
0.5 y-intercept 1
5log 1k 5k
(gradient)2n
Show
Logarithms Revision
Back NextQuit
Before a forest fire was brought under control, the spread of fire was described by a law of the form
0ktA A e
where
0A is the area covered by the fire when it was first detectedand A is the area covered by the fire t hours later.
If it takes one and a half hours for the area of the forest fire to double, find the value of the constant k.
1.50 02 kA A e 1.52 ke log 2 1.5 loge ek e
log 2 1.5e k log 2
1.5ek (2 dp)0.462.. 0.46k
Show
Logarithms Revision
Back NextQuit
The results of an experimentgive rise to the graph shown.
a) Write down the equation of the line in terms of P and Q.
It is given that
logeP p and logeQ q
bp aq stating the values of a and b.
b) Show that p and q satisfy a relationship of the form
log log be ep aq
log log loge e ep a b q
logeP a bQ Gradient 0.6 y-intercept 1.8
0.6 1.8P Q 1.8 6.0o .8 5l g 1e a ea a
0.6b
Show
Logarithms Revision
Back NextQuit
The diagram shows part of the graph of
log ( )by x a
.Determine the values of a and b.
...(1)1 log (7 )b a Use (7, 1)
...(2)0 log (3 )b a Use (3, 0)
Hence, from (2) 3 1a 2a
and from (1) 5b1 log 5b
Show
Logarithms Revision
Back NextQuit
The diagram shows a sketch ofpart of the graph of
2log ( )y x
a) State the values of a and b.
b) Sketch the graph of 2log ( 1) 3y x
1a 3b
Graph moves1 unit to the left and 3 units down
Show
Logarithms Revision
Back NextQuit
a) i) Sketch the graph of 1, 2xy a a
ii) On the same diagram, sketch the graph of 1, 2xy a a
b) Prove that the graphs intersect at a point
where the x-coordinate is 1
log1a a
11x xa a 11 x xa a 1 1xa a
log 1 log log 1xa a aa a 0 log 1ax a
log 1ax a 1log 1ax a
1
log1ax
a
Show
Logarithms Revision
Back NextQuit
Part of the graph of
105 log (2 10)y x is shown in the diagram.
This graph crosses thex-axis at the point A and
the straight line 8y at the point B. Find algebraically the x co-ordinates of A and B.
108 5 log (2 10)x 10
8log (2 10)
5x 101.6 log (2 10)x
1.610 2 10x 1.610 10 2x 14.9x B (14.9, 8)
100 5 log (2 10)x 2 10 1x 4.5x A ( 4.5, 0)
Show
Logarithms Revision
Back NextQuit
The diagram is a sketch of part of
the graph of , 1xy a a a) If (1, t) and (u, 1) lie on this curve, write down the values of t and u.
b) Make a copy of this diagram and on it
sketch the graph of 2xy a
c) Find the co-ordinates of the point of intersection of 2xy a with the line 1x
a) t a 0u b)
c) 2 1y a 2y a 21, a
Show
Logarithms Revision
Back NextQuit
The diagram shows part of the graph with equation
3xy and the straight line with equation 42y
These graphs intersect at P.
Solve algebraically the equation 3 42x
and hence write down, correct to 3 decimal places, the co-ordinates of P.
10 10log 3 log 42x 10 10log 3 log 42x
10
10
log 42
log 3x 3.40217...x
P (3.402, 42)
Show
Quit
C P Dwww.maths4scotland.co.uk
© CPD 2004
