Logarithms [49 marks]
1.
[Maximum mark: 5]
Solve the equation 2
form x
= pe
q
EXN.1.SL.TZ0.2
ln x =
where p,
ln 9 + 4. Give your answer in
q ∈ Z
+
.
the
[5]
2.
[Maximum mark: 6]
Find the range of possible values of k such that e 2x
has at least one real solution.
23M.1.SL.TZ1.5
+ ln k = 3e
x
[6]
3.
[Maximum mark: 7]
Solve the simultaneous equations
19M.1.AHL.TZ2.H_7
log 2 6x = 1 + 2 log 2 y
1 + log 6 x = log 6 (15y − 25).
[7]
4.
[Maximum mark: 5]
Solve the equation log 2 (x + 3) + log 2 (x − 3)
= 4.
17N.1.AHL.TZ0.H_1
[5]
5.
[Maximum mark: 7]
(a)
Show that log r
18M.1.AHL.TZ2.H_11
2
x =
1
2
log r x where r, x ∈ R
It is given that log 2 y + log 4 x + log 4 2x
(c)
+
.
[2]
= 0.
The region R, is bounded by the graph of the function found in
part (b), the x-axis, and the lines x = 1 and x = α where
α > 1. The area
of R is √2.
Find the value of α.
[5]
6.
[Maximum mark: 14]
EXN.2.SL.TZ0.9
The temperature T °C of water t minutes after being poured into a cup can be
modelled by T
= T0 e
−kt
where t
≥ 0 and T 0 , k are positive constants.
The water is initially boiling at 100 °C. When t
water is 70 °C.
= 10, the temperature of
(a)
Show that T 0
= 100.
(b)
Show that k
(c)
Find the temperature of the water when t
(d)
Sketch the graph of T versus t, clearly indicating any
asymptotes with their equations and stating the coordinates of
=
1
10
ln
the
[1]
10
7
.
[3]
= 15.
[2]
(e)
any points of intersection with the axes.
[4]
Find the time taken for the water to have a temperature of
50 °C. Give your answer correct to the nearest second.
[4]
7.
[Maximum mark: 5]
Solve the equation log 3
21N.1.AHL.TZ0.3
√x =
1
2 log 2 3
+ log 3 (4x ), where
3
x > 0.
[5]
© International Baccalaureate Organization, 2023