Created by T. Madas
LOGARITHMS
EXAM
QUESTIONS
Created by T. Madas
Created by T. Madas
Question 1
(**)
Show clearly that
log a 36 + 1 log a 256 − 2log a 48 = − log a 4 .
2
proof
Question 2
(**)
Simplify
log 2 5 + log 2 1.6 ,
giving the final answer as an integer.
3
Question 3
(**+)
Given that x = 2 p and y = 4q , show clearly that
log 2 ( x3 y ) = 3 p + 2q .
proof
Created by T. Madas
Created by T. Madas
Question 4
(**+)
Simplify each of the following expressions, giving the final answer as an integer.
a) log 2 3 − log 2 24 .
1
b) log a a 2 − 4log a   , a > 0 , a ≠ 1 .
a
Full workings, justifying every step, must support each answer.
−3 , 6
Question 5
(**+)
Given that y = log 2 x , write each of the following expressions in terms of y .
a) log 2 x 2
( )
b) log 2 8x 2
2y , 3+ 2y
Created by T. Madas
Created by T. Madas
Question 6
(**+)
Given that y = 4 × 102 x express x in terms of y , giving an exact simplified answer in
terms of logarithms base 10.
( )
x = 1 log10 1 y
2
4
Question 7
(**+)
An exponential curve has equation
y = ab x , x ∈ ,
where a and b are non zero constants.
Make x the subject of the above equation, giving the final answer in terms of
logarithms base 10 .
x=
Created by T. Madas
log y − log a
log b
Created by T. Madas
Question 8
(**+)
Solve the following logarithmic equation
2 log10 x + log10 3 = log10 75 .
x = 5, x ≠ −5
Question 9
(**+)
Solve the following logarithmic equation
log a x + log a ( x − 3) = log a 10 .
C2H , x = 5, x ≠ −2
Created by T. Madas
Created by T. Madas
Question 10
(***)
An exponential curve C has equation
y=
1
3x
, x∈.
a) Sketch the graph of C .
b) Solve the equation y = 2 , giving the answer correct to 3 significant figures.
3
C2B , 0.369
Created by T. Madas
Created by T. Madas
Question 11
(***)
Given that
p = log a 4
and
q = log a 5 ,
express each of the following logarithms in terms of p and q .
a) log a 100
b) log a 0.4
The final answers may not contain any logarithms.
C2G , p + 2q , 1 p − q
2
Question 12
(***)
Solve the following logarithmic equation
log 5 (4t + 7) − log 5 t = 2 .
C2L , t = 1
3
Created by T. Madas
Created by T. Madas
Question 13
(***)
Given that
p = log 2 3
and
q = log 2 5 ,
express each of the following logarithms in terms of p and q .
a) log 2 45
b) log 2 0.3
The final answers may not contain any logarithms.
MP1-N , 2 p + q , p − q − 1
Created by T. Madas
Created by T. Madas
Question 14
(***)
Solve each of the following equations, giving the final answers correct to three
significant figures, where appropriate.
a) 7 x = 10 .
b) log 2 y =
9
.
log 2 y
C2J , x ≈ 1.18 , y = 1 , 8
8
Question 15
(***)
Solve the following logarithmic equation for x .
log a ( x 2 − 10) − log a x = 2log a 3 .
x = 10, x ≠ −1
Created by T. Madas
Created by T. Madas
Question 16
(***)
Solve the following logarithmic equation for x .
2 log a x = log a 18 + log a ( x − 4) .
C2C , x = 6, 12
Question 17
(***)
Solve the following logarithmic equation
log 2 (2 z + 1) = 2 + log 2 z .
C2O , z = 1
2
Created by T. Madas
Created by T. Madas
Question 18
(***)
Solve the following logarithmic equation for y .
2 log a y − log a (5 y − 24) = log a 4 .
x = 8, 12
Question 19
(***)
It is given that x satisfies the logarithmic equation
log a x = 2(log a k − log a 2) ,
where k > 0 , a > 0 , a ≠ 1 .
a) Find x in terms of k , giving the answer in a form not involving logarithms.
Suppose instead that x satisfies
log x ( 5 y + 1) = 4 + log x 3
where x > 0 , x ≠ 1 and y > 0 , y ≠ 1 .
b) Solve the above equation expressing y in terms of x , giving the answer in a
form not involving logarithms.
x=
Created by T. Madas
k2
3x 4 − 1
, y=
4
5
Created by T. Madas
Question 20
(***)
Solve the following logarithmic equation
log 5 (125 x) = 4 .
x=5
Question 21
(***)
Solve the following logarithmic equation
1 + 2log5 x = log5 (16 x − 3) .
x = 3, x = 1
5
Created by T. Madas
Created by T. Madas
Question 22
(***)
Every £1 invested in a saving scheme gains interest at the rate of 5% per annum so
that the total value of this £1 investment after t years is £ y .
This is modelled by the equation
y = 1.05t , t ≥ 0 .
Find after how many years the investment will double.
t ≈ 14.2
Question 23
(***)
Solve each of the following logarithmic equations.
a) log x 16 = log x 9 + 2 .
b) log y 27 = 3 + log y 8 .
x = 4, x ≠ −4 , y = 3
3
3
2
Created by T. Madas
Created by T. Madas
Question 24
(***)
Solve each of the following equations, giving the final answers correct to three
significant figures, where appropriate.
a) 2 × 3x = 900 .
b) log 2 ( 7 y − 1) = 3 + log 2 ( y − 1) .
C2P , x ≈ 5.56 , y = 7
Question 25
(***+)
Simplify fully
1 + 2 log n 3 + log n 4 ,
giving the final answer as a single logarithm.
log n (36n)
Created by T. Madas
Created by T. Madas
Question 26
(***+)
Solve each of the following exponential equations, giving the final answers correct to
3 significant figures.
a) 52 x−1 = 4300 .
b)
2 y +1 =
10
2y
.
C2M , x ≈ 130 , y ≈ 1.16
Question 27
(***+)
Solve the following logarithmic equation
log 2 ( w2 + 4 w + 3) = 4 + log 2 ( w2 + w) , w ≠ −1 .
w= 1
5
Created by T. Madas
Created by T. Madas
Question 28
(***+)
Solve the following exponential equation
x
1 1
=  ,
6 2
giving the answer as single logarithm of base 2.
x = log 2 6 = 1 + log 2 3
Question 29
(***+)
Solve the following simultaneous logarithmic equations
( )
log 2 xy 2 = 0
( )
log 2 x 2 y = 3 .
C2U , x = 4, y = 1
2
Created by T. Madas
Created by T. Madas
Question 30
(***+)
Solve the following logarithmic equation
2 log 3 t = 1 + log 3 7t .
t = 21, t ≠ 0
Question 31
(***+)
Solve the following logarithmic equation
log 3 8 − 3log3 t = 3 .
C2E , t = 2
3
Created by T. Madas
Created by T. Madas
Question 32
(***+)
Solve the following logarithmic equation
log 5 (4 − w) − 2log5 w = 1 .
C2A , w = 4 , w ≠ −1
5
Question 33
(***+)
Simplify fully the following logarithmic expression, showing clearly all the workings.
(
)
(
) (
)
log 10 + 3 10 + log 10 + 90 + 90 + log 10 − 90 + 90 .
1
Created by T. Madas
Created by T. Madas
Question 34
(***+)
Solve the following logarithmic equation
log 2 y + log 2 (3 y + 4) = 2log 2 (3 y − 4) .
y = 4, y ≠ 2
3
Question 35
(***+)
Solve the following logarithmic equation
log 2 (6 − x) = 3 − log 2 x .
x = 2, 4
Created by T. Madas
Created by T. Madas
Question 36
(***+)
Solve the following logarithmic equation
log 4 x = log 3 9 .
x = 16
Question 37
(***+)
Solve each of the following equations.
1 x+ 2
a) 2 × 3 2
= 23.43 .
b) log5 ( y + 2 ) + log5 ( 4 y − 3) = 2log5 ( 2 y + 1) .
x ≈ 0.480 , y = 7
Created by T. Madas
Created by T. Madas
Question 38
(***+)
The population P of a certain town in time t years is modelled by the equation
P = A × 10kt , t ≥ 0 ,
where A and k are non zero constants.
When t = 3 , P = 19000 and when t = 6 , P = 38000 .
Find the value of A and the value of k , correct to 2 significant figures.
C2V , A = 9500, k = 0.10
Question 39
(***+)
Solve the following logarithmic equation
2 log 3 x − log 3 ( x − 2) = 2 .
C2F , x = 3, 6
Created by T. Madas
Created by T. Madas
Question 40
(***+)
Solve the following logarithmic equation
log 2 42 x = log 3 27 x−1 .
x = −3
Question 41
(***+)
Given that a ≠ 0 , b ≠ 0 , y ≠ 0 and
( )
2 + log a b + 3log a y = 2log a a 2 y ,
express y in terms of a and b , in a form not involving logarithms.
a2
y=
b
Created by T. Madas
Created by T. Madas
Question 42
(***+)
 x
2 log   − 1 = log(10 x 2 y ) , x ≠ 0 , y ≠ 0 .
 y
Find the exact value of y .
y=
Question 43
3
1
100
(***+) non calculator
The points P and Q lie on the curve with equation
y = 6 log 2 x − log 2 7 , x > 0 .
The x coordinates of P and Q are 3 and 6 , respectively.
Find the gradient of the straight line segment PQ .
2
Created by T. Madas
Created by T. Madas
Question 44
(***+)
y = 3× 2x .
a) Describe the geometric transformation which maps the graph of the curve with
equation y = 2 x , onto the graph of the curve with equation y = 3 × 2 x .
b) Sketch the graph of y = 3 × 2 x .
The curve with equation y = 2− x intersects the curve with equation y = 3 × 2 x at the
point P .
c) Determine, correct to 3 decimal places, the x coordinate of P .
C2R , vertical stretch by scale factor 3 , x ≈ −0.792
Created by T. Madas
Created by T. Madas
Question 45
(***+)
It is given that p = log 6 25 and q = log 6 2 .
Express in terms of p and q each of the following expressions
a) log 6 200
b) log 6 3.2
c) log 6 75
MP1-F , log 6 200 = p + 3q , log 6 3.2 = − 1 p + 4q , log 6 75 = p − q + 1
2
Created by T. Madas
Created by T. Madas
Question 46
(****)
y = 3x −1 , x ∈ .
a) Sketch the graph of y = 3x −1 showing the coordinates of all intercepts with the
coordinate axes.
b) Find to 3 significant figures the x coordinate of the point where the curve
y = 3x −1 intersects with the straight line with equation y = 10 .
c) Determine to 3 significant figures the x coordinate of the point where the
curve y = 3x −1 intersects with the curve y = 2 x .
3.10 , 2.71
Question 47
(****)
Solve the following logarithmic equation
16 log 2 x + 4 log 4 x + 2 log16 x = 37 , x > 0 .
x=4
Created by T. Madas
Created by T. Madas
Question 48
(****)
In 1970 the average weekly pay of footballers in a certain club was £100 .
The average weekly pay, £P , is modelled by the equation
P = A × bt ,
where t is the number of years since 1970 , and A and b are positive constants.
In 1991 the average weekly pay of footballers in the same club had risen to £740 .
a) Find the value of A and show that b = 1.10 , correct to three significant figures.
b) Determine the year when the average weekly pay of footballers in this club
will first exceed £10000 .
C2Q , A = 100 , 2019
Created by T. Madas
Created by T. Madas
Question 49
(****)
Solve each of the following equations, giving the final answers correct to three
significant figures, where appropriate.
a) 63 x + 2 = 30 .
b) log 4 (12 y + 5 ) − log 4 (1 − y ) = 2 .
c) 82t − 8t − 6 = 0 .
x ≈ −0.0339 , y = 11 ≈ 0.393 , t ≈ 0.528
28
Created by T. Madas
Created by T. Madas
Question 50
(****)
Solve the following simultaneous equations, giving your answers as exact fractions
log 2 y = log 2 x + 4
8 y = 42 x +3 .
x = 3 , y = 24
22
11
Question 51
(****)
Show clearly that
log 5 6 + 2log 5 2 − log 25 9 = 3log 5 2 .
proof
Created by T. Madas
Created by T. Madas
Question 52
(****)
Solve the following logarithmic equation
log10 ( x + 4) + log10 ( x + 16) = 1 + 2 log10 x .
x = 4, x ≠ − 16
9
Question 53
(****)
Simplify
log 4 8 − log 27 3 ,
giving the final answer as a simplified fraction.
7
6
Created by T. Madas
Created by T. Madas
Question 54
(****)
Solve each of the following equations.
( )
a) 6 × 1
2
x−4
3
= 1.89 .
b) log 2 ( 8 y − 1) − 2log 2 ( y + 1) = 3 − log 2 ( y + 4 ) .
C2D , x ≈ 9.00 , y = 4
5
Question 55
(****)
Simplify
1
log 1 8 + log 2 ,
2
8
giving the final answer as an integer.
−6
Created by T. Madas
Created by T. Madas
Question 56
(****)
Given that a = log b 16 , express log b (8b) in terms of a .
1+ 3 a
4
Question 57
(****)
log a y = 1 and log8 a = x + 1 .
3
Show clearly that y = 2 x +1
proof
Created by T. Madas
Created by T. Madas
Question 58
(****)
It is given that
p = log 6 25
and
q = log 6 2 .
Simplify each of the following logarithms, giving the final answers in terms of p , q
and positive integers, where appropriate.
i.
log 6 ( 200 ) .
ii.
log 6 ( 3.2 ) .
iii.
log 6 ( 75) .
SYN-B , p + 3q , 4q − 1 p , p + 1 − q
2
Created by T. Madas
Created by T. Madas
Question 59
(****)
Solve the following exponential equation, giving the answer correct to 3 s.f.
22 x − 2 x − 6 = 0 .
x ≈ 1.58
Question 60
(****)
Two curves C1 and C2 are defined for all values of x and have respective equations
y1 = 8 x
and
y2 = 2 × 3x .
Show that the x coordinate of the point of intersection of the two curves is given by
1
.
3 − log 2 3
proof
Created by T. Madas
Created by T. Madas
Question 61
(****)
Solve the following logarithmic equation
log 2 x = log 4 2 .
x= 2
Created by T. Madas
Created by T. Madas
Question 62
(****)
The functions f and g are defined as
( )
f ( x ) = 3 2− x − 1 , x ∈ , x ≥ 0
g ( x ) = log 2 x , x ∈ , x ≥ 1 .
a) Sketch the graph of f .
•
Mark clearly the exact coordinates of any points where the curve meets
the coordinate axes. Give the answers, where appropriate, in exact
form in terms of logarithms base 2 .
•
Mark and label the equation of the asymptote to the curve.
b) State the range of f .
c) Find f ( g ( x ) ) in its simplest form.
SYN-B , ( 0, 2 ) , ( log 2 3, 0 ) , y = −1 , −1 < f ( x ) ≤ 2 , f ( g ( x ) ) =
Created by T. Madas
6
−1
x
Created by T. Madas
Question 63
(****)
Solve the following logarithmic equation
log 3 x = log9 27 .
x=3 3
Question 64
The points
(****)
( 2,10 )
and ( 6,100 ) lie on the curve with equation
y = ax n ,
where a and n are non zero constants.
Find, to three decimal places, the value of a and the value of n .
MP1-H , a = 2.339 , n ≈ 2.096
Created by T. Madas
Created by T. Madas
Question 65
(****)
Solve the following exponential equation, giving the answer correct to 3 s.f.
( )
4 y − 3 2 y − 10 = 0 .
C2N , y ≈ 2.32
Question 66
(****)
Solve the following logarithmic equation
log 3 x 2 − log 9 x = 3 .
x = 9, x ≠ 0
Created by T. Madas
Created by T. Madas
Question 67
(****)
Show that x = 4 and y = 8 is the only solution pair of the following logarithmic
simultaneous equations
log 2 (3 x + 4) = 1 + log 2 y .
2 log 2 y = 3log 2 x .
proof
Created by T. Madas
Created by T. Madas
Question 68
(****)
A population P of an endangered species of animals was introduced to a park.
The population obeys the equation
P=
125kat
k + 2a t
, t ≥ 0,
where k and a are positive constants, and t is the time in years since the species was
introduced to the park .
Initially 100 individual animals were introduced to the park, and this population
doubled after 5 years.
a) Show that k = 8 .
b) Find the value of a , correct to 4 significant figures.
c) Determine the value of t when the P = 400 .
d) Explain why this population cannot exceed 500 .
MP1-C , a ≈ 1.217 , t ≈ 14.13
Created by T. Madas
Created by T. Madas
Question 69
(****)
Solve the following logarithmic equation
log 3 x − log 9 2 x = 2 .
x = 162, x ≠ 0
Question 70
(****)
Solve the following logarithmic equation
log 4 x − log16 ( x − 4 ) = 1 .
C2I , x = 8
Created by T. Madas
Created by T. Madas
Question 71
(****)
Solve the following simultaneous equations
log 2 x + 2 log 4 y = 4
x + y = 10
MP1-G , x = 2, y = 8
Question 72
or x = 8, y = 2
(****)
Given that a is positive constant greater than 1 , solve the following logarithmic
equation
log a x = log a 2 ( x + 20 ) .
SYN-P , x = 5
Created by T. Madas
Created by T. Madas
Question 73
(****)
Two curves C1 and C2 are defined for all values of x and have respective equations
y1 = 7 x and
y2 = 2 × 5 x .
Show that the x coordinate of the point of intersection of the two curves is given by
1
.
log 2 7 − log 2 5
SYN-G , proof
Question 74
(****)
Solve the following exponential equation, giving the answer correct to 3 s.f.
3t +1 = 6 + 32t −1
SYN-I , t = 1 or t ≈ 1.63
Created by T. Madas
Created by T. Madas
Question 75
(****)
Solve the following simultaneous logarithmic equations
log y x = 5
log 2 x = 2 + log 2 y .
Give the answer as exact simplified surds.
C2K , x = 4 2, y = 2
Question 76
(****)
Solve the following logarithmic equation
log 4 x + log x 16 = 3 , x > 0 , x ≠ 1 .
x = 4, 16
Created by T. Madas
Created by T. Madas
Question 77
(****)
Find, to the nearest integer, the solution of the following exponential equation
1 × 42 x = 500500 .
2
x ≈ 1121
Question 78
(****)
Solve the following simultaneous logarithmic equations.
3log8 ( xy ) = 4log 2 x
log 2 y = 1 + log 2 x
SYN-C , x = 2, y = 8
Created by T. Madas
Created by T. Madas
Question 79
(****)
y = log 2 x
y
P
y = log 2 ( x − 4 )
Q
O
x
A
B
x=k
The figure above shows the graphs of the curves with equations
y = log 2 x ,
and y = log 2 ( x − 4 ) .
The points A and B are the respective x intercepts of the two graphs.
a) Describe the geometric transformation which maps the graph of y = log 2 x
onto the graph of y = log 2 ( x − 4 ) .
b) State the distance AB .
The straight line with equation x = k , where k is a positive constant, meets the graph
of y = log 2 x at the point P and the graph of y = log 2 ( x − 4 ) at the point Q .
c) Given that the distance PQ is 2 units determine the value of k .
C2V , translation, 4 units to the "right" , AB = 4 , k = 16
3
Created by T. Madas
Created by T. Madas
Question 80
(****)
The radioactive decay of a phosphorus isotope is modelled by the equation
m = m0 × 2−0.2t , t ≥ 0
where m is the mass of phosphorus left, in grams, and t is the time in days since the
decay started. The initial mass of phosphorus is m0 .
a) Find the mass of the phosphorus left, when an initial mass of 20 grams is left
to decay for 10 days, according to this model.
An initial mass, m0 grams, of this type of phosphorus decays to
m0
grams in T days.
64
b) Find the value of T .
After N days have elapsed, less than 1% of this type of phosphorus remains from its
initial mass m0 .
c) Find the smallest integer value of N .
C2X , m = 5 , T = 30 , N = 34
Created by T. Madas
Created by T. Madas
Question 81
(****)
Solve the following logarithmic equation
log 4 x − 2log x 4 = 1 .
C2Z , x = 1 , 16
4
Question 82
(****)
Solve the following simultaneous logarithmic equations.
log 2 ( y − 1) = 1 + log 2 x
2 log 3 y = 2 + log3 x .
MP1-A , x = 1, y = 3 or x = 1 , y = 3
4
2
Created by T. Madas
Created by T. Madas
Question 83
(****)
Solve the following logarithmic equation
log 2 128 − log 2 8
= log 2 x .
log 2 x
C2V , x = 4, 1
4
Question 84
(****)
Two curves C1 and C2 are defined for all values of x and have respective equations
y1 = 9 x
and
y2 = 6 × 5 x .
Show that the x coordinate of the point of intersection of the two curves is given by :
1 + log 3 2
2 − log 3 5
MP1-K , proof
Created by T. Madas
Created by T. Madas
Question 85
(****+)
Solve each of the following equations.
a)
1 × 43 x +1 = 600600 .
2
b) log3 ( 2 y + 5) = 1 − log3 y .
MP1-I , x = 922.7152024... , y = 1
2
Created by T. Madas
Created by T. Madas
Question 86
(****+)
( )
y= 1
2
x
.
a) Describe the geometric transformation which maps the graph of the curve with
x
equation y = 2 x , onto the graph of the curve with equation y = 1 .
2
( )
b) Sketch the graph of y = 3 × 2 x .
( )
The curve with equation y = 1
2
point P .
x
intersects the curve with equation y = 3 × 2 x at the
c) Determine, as an exact simplified surd, the y coordinate of P .
C2Z , reflection in the y axis , y = 3
Created by T. Madas
Created by T. Madas
Question 87
(****+)
Solve each of the following logarithmic equations, giving the answers in exact
simplified form where appropriate.
a)
1 
log 2 (256 x 2 ) = 1 + 2log 2  x 4  .
2 
 y
b) 2 log 2   + log 2 y = 8 .
2
SYN-E , x = ± 8 , y = 16
Created by T. Madas
Created by T. Madas
Question 88
(****+)
Solve the following simultaneous logarithmic equations.
( )
log 2 x 2 y = 2
11 + 1 log 2 y = 3log 2 x .
2
MP1-R , x = 8, y = 1
16
Created by T. Madas
Created by T. Madas
Question 89
(****+)
Solve each of the following equations, giving the final answers correct to three
significant figures, where appropriate.
a) 4 × 3x + 2 = 3 × 4 x .
(
)
(
)
b) log a 1 + x = 1 log a 9 + 16 x .
2
MP1-E , x ≈ 8.64 , y = 16
Created by T. Madas
Created by T. Madas
Question 90
(****+)
Solve the following exponential equation.
4 x+1 × 31−2 x = 24 .
Give the answer correct to 3 decimal places.
SYN-K , x ≈ −0.855
Created by T. Madas
Created by T. Madas
Question 91
(****+)
Solve the following logarithmic equation.
2log 2 x + log 2 ( x − 1) − log 2 ( 5 x + 4 ) = 1 .
MP1-P , x = 4
Created by T. Madas
Created by T. Madas
Question 92
(****+)
The following three numbers
log10 2 ,
(
)
log10 2 x − 1 ,
(
)
log10 2 x + 3 ,
are consecutive terms in an arithmetic progression.
Determine the value of x as an exact logarithm, of base 2 .
SP-G , x = log 2 5
Created by T. Madas
Created by T. Madas
Question 93
(****+)
f ( x ) = 24 x
Show that the solution of the equation f ( x − 1) =
x=
5
is given by
8
1
.
4log10 2
SYN-S , proof
Created by T. Madas
Created by T. Madas
Question 94
(****+)
2 log 2 x − log 2 y = 1
(
)
log 2 4 x y = 1 .
Solve the above simultaneous logarithmic equations, giving the final answers as exact
powers of 2.
MP1-X , x = 2
Created by T. Madas
− 14
, y=2
− 32
Created by T. Madas
Question 95
(****+)
Solve the following logarithmic equation
5 × 5log x + 52−log x = 30 ,
x >0.
SYN-T , x = 1, x = 10
Created by T. Madas
Created by T. Madas
Question 96
(****+)
It is given that
3

3
log a x
r
=
r =1

( log a x )r ,
r =1
where a and x are positive numbers such that x ≠ a , x ≠ 1 and a > 1 .
Show clearly that
x=a
−1± 21
2
.
SP-D , proof
Created by T. Madas
Created by T. Madas
Question 97
(****+)
2log 2 x + log 2 ( x − 1) − log 2 ( 5 x + 4 ) = 1 .
Find the only real root of the above logarithmic equation.
SYN-V , x = 4
Created by T. Madas
Created by T. Madas
Question 98
(****+)
Solve the following simultaneous logarithmic equations
y log x = 100
log
xy
= 1,
10
given further that x , y ∈ , with x > 0 , y > 0 .
SP-A , x = 10, y = 100, or the other way round
Created by T. Madas
Created by T. Madas
Question 99
(****+)
Solve the following logarithmic equation, over the largest real domain
log 2− x  2 x 2 − 1 = 2 , x ∈ .


SP-J , x = −5
Question 100
(****+)
Solve the following simultaneous equations
a 2 x × b3 y = c 5
a3 x × b2 y = c10
Give the answers in exact form in terms of log a , log b and log c .
SPX-M , x =
Created by T. Madas
4 log c
log c
, y=−
log a
log b
Created by T. Madas
Question 101
(****+)
Solve the following logarithmic equation
log 4 x 2
5 + log 4 x
2
2
+ ( log 4 x ) = 0 .
MP1-U , x = 1, 1 , 1
2 16
Created by T. Madas
Created by T. Madas
Question 102
(****+)
Solve the following equation
( a4 − 2a2b2 + b4 )
x −1
= ( a − b)
2x
( a + b )−2
Give the answers in exact form in terms of a and b .
SPX-Q , x =
Question 103
log ( b − a )
log ( b + a )
(*****)
Solve the following logarithmic equation
2 − log 4 x7
7 − log 4 x
2
2
+ ( log 4 x ) = 0
C2S , x = 2, x = 4, x = 16
Created by T. Madas
Created by T. Madas
Question 104
(*****)
Show by valid mathematical arguments that
8
8! < 9 9!
SP-T , proof
Created by T. Madas
Created by T. Madas
Question 105
(*****)
i. Simplify the following expression.
9log 24 2 + log 24 27 .
Show detailed workings in this simplification.
ii. It is given that
5 × 2t −1 = 2 × t 2t

(10k )t = k
Determine the value of k .
SPX-K , 3 , k = 1.25
Created by T. Madas
Created by T. Madas
Question 106
(*****)
On the 1st January 2000 a rare stamp was purchased at an auction for £16384 and by
the 1st January 2010 its value was four times as large as its purchase price.
The future value of this stamp, £V , t years after the 1st January 2000 is modelled by
the equation
V = Apt , t ≥ 0 ,
where A and p are positive constants.
On the 1st January 1990 a different stamp was purchased for £2.
The future value of this stamp, £U , t years after the 1st January 1990 is modelled by
the equation
U = Bqt , t ≥ 0 ,
where B and q are positive constants.
Given further that q = p 2 , determine the year when the two stamps will achieve the
same value according to their modelling equations.
SP-V , 2012
Created by T. Madas
Created by T. Madas
Question 107
(*****)
It is given that
•
a and x are positive real numbers such that x ≠ a , x ≠ 1 and a > 1 .
•
n is a positive integer such that n > 1 .
Show that the equation
n

r =1
n
r
log a x =
(
r
log a x ) ,
r =1
can be written as
n
2 ( log a x ) − n ( n + 1) log a x + ( n − 1)( n + 2 ) = 0 .
SP-X , proof
Created by T. Madas
Created by T. Madas
Question 108
(*****)
It is given that
a log b ≡ blog a , a > 0 , b > 0 .
i. Prove the validity of the above result.
ii. Hence, or otherwise, solve the following simultaneous equations
log x + 10log y = 7 ,
x y = 1012 .
SYN-X , (10000,3 ) , (1000, 4 )
Created by T. Madas
Created by T. Madas
Question 109
(*****)
The product operator
∏ , is defined as
k
∏[
ui ] = u1 × u2 × u3 × u4 × ...× uk −1 × uk .
i =1
Given that k ∈ , use a detailed method to find the value of
2k −1
∏ 
log r ( r + 1)  .
r =2
SPX-R , k
Created by T. Madas
Created by T. Madas
Question 110
(*****)
Solve the following logarithmic equation.
3 + 8log 1  8 + 4 3 − 8 − 4 3  = 0 ,


k 
k > 0 , k ≠ 1.
SP-F , k = 16
Created by T. Madas
Created by T. Madas
Question 111
(*****)
Solve the following logarithmic equation.
xlog 2 x
x2
= 8 , x ∈ ( 0, ∞ )
SPX-R , x = 1 ∪ x = 8
2
Created by T. Madas
Created by T. Madas
Question 112
(*****)
Solve the following logarithmic equation.
( )
( )
log 4 x 1 + log x 8 + log 1 x 1 = 1 , x > 0 , x ≠ 1 .
2
2
4
2
Give the answers in exact simplified form where appropriate.
1
SPX-F , x = 16, x = 2 2
Created by T. Madas
(7±
73
)
Created by T. Madas
Question 113
(*****)
Solve the following simultaneous equations.
∞

2
r =0
∞
r
[ log 2 a ]
=

k =1
1
(1 + b )
−k
and

1
(1 + b )
k =1
−k
−

[ log 2 a ]r
=
r =0
7
.
5
You may leave the answers as indices in their simplest form, where appropriate.
13


SP-W , [ a, b] =  3 , 1  =  − 4 , 2 5 
2 4  5

Created by T. Madas
Created by T. Madas
Question 114
(*****)
It is given that
a log b ≡ blog a , a > 0 , b > 0 .
a) Prove the validity of the above result.
b) Hence, or otherwise, solve the equation
3log x + 3 × x log 3 = 36 .
SP-Y , x = 100
Created by T. Madas
Created by T. Madas
Question 115
(*****)
It is given that
4 p − 1 q = log 6 ( 3.6 )
2
and
q − p + 1 = log 6 ( 75) .
Solve these simultaneous equations, to show that
p = log 6 k ,
where k is a positive integer to be found.
SPX-J , k = 2
Created by T. Madas
Created by T. Madas
Question 116
(*****)
f ( x ) ≡ 4 x +1 × 31−2 x , x ∈ .
Determine the value of f ( a ) , where a =
log10 2
.
log10 4 − log10 9
SP-I , f ( a ) = 24
Created by T. Madas
Created by T. Madas
Question 117
(*****)
The positive solution of the quadratic equation x 2 − x − 1 = 0 is denoted by φ , and is
commonly known as the golden section or golden number.
Show, with a detailed method, that the real solution of the following exponential
equation
2 x + 4x = 8x ,
can be written in exact form as
log 2 φ .
X , proof
Created by T. Madas
Created by T. Madas
Question 118
(*****)
Show that the following logarithmic equation has no real solutions.
log x2 + 2  2 x 4 − 2 x3 + 7 x 2 − 2 x + 5 = 2 , x ∈ .


SP-L , proof
Created by T. Madas
Created by T. Madas
Question 119
(*****)
It is given that for x > 0 , x ≠ 1 and y > 0 , y ≠ 1
log x y = log y x
and
log x ( x − y ) = log y ( x + y ) .
Show that
x4 − x2 − 1 = 0 .
SP-H , proof
Created by T. Madas
Created by T. Madas
Question 120
(*****)
log sin x cos x ( sin x ) × log sin x cos x ( cos x ) =
1
.
4
Show that the solution of the above equation is given by
x = 1 π ( 8n − 7 ) , n ∈ .
4
C2T , proof
Created by T. Madas