# MATHS TASK: Indices, Exponential and Logarithmic Functions, Applications

MATHS TASK- Indices, Exponential and Logarithmic Functions, Applications.docx

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PART 1
In question 1 and 2 use the index laws to simplify the given expressions, giving your answers in terms
of positive exponents (you may assume non-zero bases and integer exponents throughout).
1.
(i)
16m 7 x 5
12m 4 x 3
(ii)
(9x-1y-3)2? (3xy 6)2
(iii)
(a2b3c0)1
(iv)
x2y3
(v)
ax 2
x – 2 a -3
(vi)
18c 4 d 2
3c 5 d -2
(viii)
? a -5 ?
? 3 4?
? 3a b ?
4 2 4
(vii)
(2a b )
(ix)
9c 7 d
18c 2 d
?
2c3
4c3d 2
-1
7a -5b 2
( -2a b )
(x)
3
3
21a -3b 2
?
4a -1b
For question 2 parts (i) to (iii), you are strongly advised to convert all numbers into the product of their
prime factors before simplifying. For example 8 = 23, 81= 34 and 24 = 23?3.
For part (iv), the best approach is to let 100 = 102 before simplifying.
For parts (v) and (vi), use the rule a -1 =
1
1
a
2.
1
= and apply all simplification rules correctly.
a
(ii)
? 8 ? 35 ? 2-19
?? – 7
?? ?
? 2 ? 9 ? 243
(iii)
3b3 x ? 2b -2 y
24b 2 x – 3 y
(iv)
10 -2 n (100 n +1 ) -2
(10 -1 ) 2 n (10 2 ) -3n -1
(v)
a -1 – b -1
a -1b -1
(vi)
(a + b) -1 – b -1
b
(i)
8 ? 2 -5 ? 3-5
9 ? 2 -7 ? 81
Hint: Be careful in parts (v) and (vi). ( x + y )
3.
Show that
36 n + 4 n
= 2 n for any integer n.
n
n
18 + 2
-1
? x -1 + y -1
PART 2
1
Find the exact value of each of the following.
Show all working out carefully.
3
1
(i)
42
(iii)
64 3
5
– 15
(ii)
256 4
(iv)
252.5
(vi)
? 25 ?
? ?
?9?
-1.5
(v)
32
(vii)
81 ? (0.125) ? (0.1)
? 4- 2 ?
?? -3 ??
? 25 ?
3
2.
– 34
-2
1
3
(viii)
– 23
Simplify each of the following, expressing your answers with positive indices.
1
3
2
1
2
(i)
a ?a ?a
(iii)
u 3 u 3 + 4u
1
3
(v)
(
2
1
4
– 13
(ii)
) (iv) (x
-1
9x
1
2
1
) (
2
3
3
(vi)
– 23
– 13
1
1
+ y2 + x2 – y2
( x) ? x
27a -9b 6
? 1 3 -2 ?
? ab ?
?2
?
12 x 3 ? 2 x
)
2
– 12
-4
x 5 ? x3
Hint: In part (iv), be careful to use the expansion of perfect squares.
3.
Express in simplest surd form:
2 2 ? 5 2 (ii)
(iii)
5 x2 y 2 ? 2 x-6 y 0
(v)
(18a b )
1
1
( )
3
4 3
1
2
2
4
1
33 ? 33 ? 32
(i)
3
(iv)
18 2
4.
Solve the following equations exactly.
(i)
()
(ii)
2 x + 2 = 2 x + 12
(iii)
11(10 x ) – 10 = 102 x
1
2
-x
= 64
5.
Sketch the following pairs of graphs on the same set of axes. (One set of axes for each pair of
graphs.)
(a) y = 2x and y = 2x+2- 4
(b) y = 3xandy = -3x + 2
What transformations have to be performed to the 1st graph in order to sketch the 2nd graph?
For each of the transformed graphs, state
the domain
the range
the coordinates of the intercepts
the equation of the asymptote.
CAS Calculators
Check your graphs using your calculator. Are your sketch graphs the same as the graphs shown on
6.
Using your CAS calculator, solve the following equations using two different methods:
using the graphing screen
using the solve command
In both cases, give your answers to 2 decimal places.
1
10
(a)
2- x =
(b)
10 x – 50 = 0
Show all calculator screens with your solutions. (They can be copied by hand.)
PART 3
1.
Express each of the following in logarithmic form:
(a)
110 = 1
(b)
4-1 = 0.25
(c)
2.
3.
4.
cx = y
Express each of the following in index form:
(a)
log216 = 4
1
25
(b)
log5
= -2
(c)
logam = n
Evaluate the following:
(a)
log3 81
(b)
log7 7
(c)
log9 3
Simplify each of the following:
(a)
log10 20 + log10 5
1
4
3
4
(b)
log3 -log3
(c)
log5 100 – log5 8
(d)
log424 – log4 2 – log4 6
(e)
1 + log3 12  2log3 2
(f)
2
3
1
logaa2 + loga a
2
5.
6.
Solve each of the following logarithmic equations:
(a)
log10x = 4
(b)
loga 8 = 3
(c)
log5(2t 3) = 3
Find the inverse of each of the following functions:
(a)
y= log10 3x
(b)
7.
y = 5x
On the same pair of axes, without using a calculator,sketch the graphs of
(a)
y = log10x, and y = 10x.
(b)
y = log3x, and y = 3x.
Label each graph and your axes. Indicate the value of the intercepts.
CAS Calculators
8. Evaluate the following, giving answers correct to 3 decimal places:
(a)
log10 33
(b)
loge 61
(c)
log2 5
Include a copy of the calculator screens with your answers.
9.
(a)
Find the inverse function of each of the following functions:
y= log10 (7x) + 2
(b)
y = 23x
Include a copy of the calculator screens with your answers.
10.
3y = 7
Solve each of the following equations, giving your answers correct to 3 decimal places.
52x + 1 = 9
0.3t = 2
3x + 1 = 2x
22n  14 ? 2n + 49 = 0
Include a copy of the calculator screens with your answers.
If f ( x ) = x2 – 5x and g ( x ) = e2x , use your calculator, to find:
(a)
(b)
f ( g ( x ))
g ( f ( x ))
Include a copy of the calculator screens with your answers.
PART 4
CAS Calculators
Show ALL working out ,including calculator screens.
(The calculator screens can be copied by hand)
1
A discus thrower competes at several competitions during the year. The best distance, d metres,
that she achieves at each consecutive competition is modelled by
d = 50 + log10 (15n), where n is the competition number.
(a) Find the distance (correct to two decimal places) thrown at the
(i) 1st (ii) 3rd (iii) 6th (iv) 10th competition.
(b)
Using your calculator to help you, sketch the graph of d = 50 + log10 (15n) by
hand.
Remember to label appropriately the axes and the scale on the axes.
(Hint: you may need to alter the WINDOW of your calculator to view the graph.)
Write down the equations of any asymptotes.
(c)
2
How many competitions does it take for the thrower to reach a distance of 53metres?
A number of deer, N, are introduced to a reserve and its population can be predicted by the
modelN = 140 (1.12t ), where t is the number of years since introduction.
(a)
Find the initial number of deer in the reserve.
(b)
Find the number of deer after
(i)
1 year
(ii)
3 years
(iii)
7 years
(c)
How long does it take for the population to double?
Sketch, by hand, the graph of N versus t. Use your calculator if necessary to
help you, but remember to label the axes and the scale on the axes appropriately.
Is the model reliable for an indefinite time period? Give reasons for your answer.
3
(a) Find an exponential model of the form y = a ? bx to fit the following data. (Give your values
for a and b correct to one decimal place.)
x
y
0
2
2
5
4
13
5
20
10
200
(b) Express the model you have found in (a) in the form y = a ? 10kx
(c) Hence find an expression for x in terms of y.
4
Complete this application problem.

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