Mathematics · Oct / Nov 2022 · Paper 21 · Question 10
YMA01
As printed in the paper
The question as text
10. Given a = log2 3
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Show the answer and mark scheme
10(i)[3]
2a
(b)
3
=
1
log
log
3
−
log 16,
=
2 a
−
4
2
16
2
2
(i)(a)
B1: 2a
(b)
3
=
M1: Uses the subtraction rule for logs to write
log
log
3
−
log 16
seen or implied.
2
16
2
2
A1: 1
2 a −
4
Do not isw here so if they reach 1
2 a −
4
a
8
correctly and then say
=
−
score M1A0
(ii) This general guidance will apply to most cases you will come across:
M1: Takes logs (same base) of both sides and applies the addition rule for logs on the lhs
See possible rearrangements below which may be seen before the addition rule is applied.
dM1: Correct method to make x the subject. Condone slips in rearrangement but there must be
more than one term in x. Award for collecting terms in x on one side and non x terms on the
other side, factorising and then dividing. Dependent on the first method mark.
B1: The correct power law seen for logs. Allow this mark to score independently so sight of
(
)
log 2
x
=
x
log 2, log 3
x
=
x
log 3, log 6
x
=
x
log 6, log 2
x
+
4
=
x
+
4 log 2
e.g.
etc. can
2
2
2
2
2
2
2
2
+ =
score this mark. (Condone e.g.
log 2
x
4
x
+
4log 2
for this mark)
2
2
Note that this may follow incorrect work e.g.
(
)
3
x
×
2
x
+
4
=
6
log 3
x
×
log 2
x
+
4
=
log 6
x
log 3
×
x
+
4 log 2
x
+
4
=
log 6
2
2
2
2
2
2
a
−
3
or e.g. 3
−
a
A1:
−−
1
a
+
a
1
Note that candidates are unlikely to be working in anything other than base 2 so you can condone
the omission of the base throughout but see 3rd below the main scheme below.
3
a
−
Note that for reference,
= −
0.54741122...
which may be useful in some circumstances.
a
+
1
Examples:
3
x
×
2
x
+
4
=
6
log 3
x
+
log 2
x
+
4
=
log 6
2
2
2
or
3
x
2
x
+
4
6
3
x
2
x
2
4
6
log 3
x
log 2
x
log 2
4
log 6
×
=
×
×
=
+
+
=
2
2
2
2
or
3
x
2
x
4
6
3
x
2
x
3
3
log 3
x
log 2
x
3
log 3
×
+
=
×
+
=
+
+
=
2
2
2
or
3
3
3
x
×
2
x
+
4
=
6
3
x
×
2
x
=
log 3
x
+
log 2
x
=
log
8
2
2
2
8
or
3
x
×
2
x
+
4
=
6
3
x
−
1
×
2
x
+
3
=
1
log 3
x
−
1
+
log 2
x
+
3
=
log 1
2
2
2
Examples:
(
)
x
log 3
+
x
+
4 log 2
=
log 6
x
(log 3
+
log 2)
=
log 6
−
4
x
=
...
2
2
2
2
2
2
or
x
log 3
x
log 2
4
log 6
x
(
log 3
log 2
)
log 6
4
x
...
+
+
=
+
=
−
=
2
2
2
2
2
2
or
(
)
x
log 3
+
x
+
3 log 2
=
log 3
x
(log 3
+
log 2)
=
log 3
−
3log 2
x
=
...
2
2
2
2
2
2
2
or
3
3
(
)
x
log 3
+
x
log 2
=
log
x
log 3
+
log 2
=
log
x
=
...
2
2
2
2
2
2
8
8
or
(
1 log 3
)
(
3 log 2
)
0
3
...
x
−
+
x
+
=
ax
+
x
=
a
−
x
=
2
2
log
a
b
=
b
log
a
2
2
a
−
3
x
+
=
a
1
Some alternatives you may come across are below:
Alternative not requiring the addition law:
3
3
3
3
x
×
2
x
+
4
=
6
3
x
×
2
x
=
log 3
x
×
2
x
=
log
log6
x
=
log
8
2
2
8
2
8
3
3
log6
x
=
log
x
log 6
=
log
x
=
...
2
8
2
2
8
3
log
log 3
log 8
3
2
8
−
a
−
x
=
=
2
2
=
log 6
log 6
a
+
1
2
2
Score as:
M1: Divides by 24, takes logs of both sides and writes 3
x
×
2
x
as 6x
B1:
log 6
x
=
x
log 6
2
2
dM1: Makes x the subject
a
−
3
or e.g. 3
−
a
A1:
−−
a
+
1
a
1
Alternative not requiring logs:
a =
log 3
3
=
2a
2
4
×
2
ax
×
2
x
+
= ×
3 2
=
2
a
2
(
)
2
ax x
+ +
4
=
2
a
+
1
ax
+
x
+
4
=
a
+
1
x a
+
1
=
a
−
3
x
=
...
a
−
3
x
=
+
a
1
Score as:
B1:
a =
log 3
3
=
2a
2
M1: Attempts to write all terms as powers of 2
dM1: Combines and equates powers and makes x the subject as in main scheme
a
−
3
or e.g. 3
−
a
A1:
−−
a
+
1
a
1
Alternative using change of base:
3
x
×
2
x
+
4
=
6
2
x
+
3
=
3
1
−
x
log2
x
+
3
=
log3
1
−
x
(
)
(
)
x
+
3 log2
=
1
−
x
log3
log 2
log 3
(
3
)
(
1
)
3
(
1
)
(
1
)
3
...
x
+
2
=
−
x
2
x
+
=
a
−
x
x a
+
=
a
−
x
=
log 10
log 10
2
2
Score as:
M1: Divides by 2 and 3x and takes logs of both sides
+ =
(
)
(
)
B1: E.g.
log2
x
3
x
+
3 log2
or
log3
1
−=
x
1
−
x
log3
dM1: Changes to base 2 correctly and makes x the subject as main scheme
a
−
3
or e.g. 3
−
a
A1:
−−
1
a
+
a
1
Alternative using logs base 6:
3
3
3
3
x
2
x
+
4
6
3
x
2
x
6
x
log 6
x
log
×
=
×
=
=
=
6
6
8
8
8
3
3
log 6
x
=
log
x
log 6
=
log
6
6
8
6
6
8
3
log
3
2
8
x =
log 8
=
6
log 6
2
3
log
log 3
−
log 8
a
−
3
2
8
=
2
2
=
log 6
log 3
+
log 2
a
+
1
2
2
2
as 6x and takes logs base 6 of both sides
M1: Divides by 24, writes 3
x
×
2
x
3
3
which may be implied by e.g.
6
x
x
log
=
=
6
8
8
B1: For
log 6
x
=
x
log 6
(may be implied)
6
6
dM1: Changes to log base 2 correctly (and makes x the subject)
A1: Correct expression
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with its registered office at 80 Strand, London, WC2R 0RL, United Kingdom
10(ii)[4]
Examples:
3
x
×
2
x
+
4
=
6
log 3
x
+
log 2
x
+
4
=
log 6
2
2
2
or
3
x
2
x
+
4
6
3
x
2
x
2
4
6
log 3
x
log 2
x
log 2
4
log 6
×
=
×
×
=
+
+
=
2
2
2
2
or
3
x
2
x
4
6
3
x
2
x
3
3
log 3
x
log 2
x
3
log 3
×
+
=
×
+
=
+
+
=
2
2
2
or
3
3
3
x
×
2
x
+
4
=
6
3
x
×
2
x
=
log 3
x
+
log 2
x
=
log
8
2
2
2
8
or
3
x
×
2
x
+
4
=
6
3
x
−
1
×
2
x
+
3
=
1
log 3
x
−
1
+
log 2
x
+
3
=
log 1
2
2
2
Examples:
(
)
x
log 3
+
x
+
4 log 2
=
log 6
x
(log 3
+
log 2)
=
log 6
−
4
x
=
...
2
2
2
2
2
2
or
x
log 3
x
log 2
4
log 6
x
(
log 3
log 2
)
log 6
4
x
...
+
+
=
+
=
−
=
2
2
2
2
2
2
or
(
)
x
log 3
+
x
+
3 log 2
=
log 3
x
(log 3
+
log 2)
=
log 3
−
3log 2
x
=
...
2
2
2
2
2
2
2
or
3
3
(
)
x
log 3
+
x
log 2
=
log
x
log 3
+
log 2
=
log
x
=
...
2
2
2
2
2
2
8
8
or
(
1 log 3
)
(
3 log 2
)
0
3
...
x
−
+
x
+
=
ax
+
x
=
a
−
x
=
2
2
log
a
b
=
b
log
a
2
2
a
−
3
x
+
=
a
1
Some alternatives you may come across are below:
Alternative not requiring the addition law:
3
3
3
3
x
×
2
x
+
4
=
6
3
x
×
2
x
=
log 3
x
×
2
x
=
log
log6
x
=
log
8
2
2
8
2
8
3
3
log6
x
=
log
x
log 6
=
log
x
=
...
2
8
2
2
8
3
log
log 3
log 8
3
2
8
−
a
−
x
=
=
2
2
=
log 6
log 6
a
+
1
2
2
Score as:
M1: Divides by 24, takes logs of both sides and writes 3
x
×
2
x
as 6x
B1:
log 6
x
=
x
log 6
2
2
dM1: Makes x the subject
a
−
3
or e.g. 3
−
a
A1:
−−
a
+
1
a
1
Alternative not requiring logs:
a =
log 3
3
=
2a
2
4
×
2
ax
×
2
x
+
= ×
3 2
=
2
a
2
(
)
2
ax x
+ +
4
=
2
a
+
1
ax
+
x
+
4
=
a
+
1
x a
+
1
=
a
−
3
x
=
...
a
−
3
x
=
+
a
1
Score as:
B1:
a =
log 3
3
=
2a
2
M1: Attempts to write all terms as powers of 2
dM1: Combines and equates powers and makes x the subject as in main scheme
a
−
3
or e.g. 3
−
a
A1:
−−
a
+
1
a
1
Alternative using change of base:
3
x
×
2
x
+
4
=
6
2
x
+
3
=
3
1
−
x
log2
x
+
3
=
log3
1
−
x
(
)
(
)
x
+
3 log2
=
1
−
x
log3
log 2
log 3
(
3
)
(
1
)
3
(
1
)
(
1
)
3
...
x
+
2
=
−
x
2
x
+
=
a
−
x
x a
+
=
a
−
x
=
log 10
log 10
2
2
Score as:
M1: Divides by 2 and 3x and takes logs of both sides
+ =
(
)
(
)
B1: E.g.
log2
x
3
x
+
3 log2
or
log3
1
−=
x
1
−
x
log3
dM1: Changes to base 2 correctly and makes x the subject as main scheme
a
−
3
or e.g. 3
−
a
A1:
−−
1
a
+
a
1
Alternative using logs base 6:
3
3
3
3
x
2
x
+
4
6
3
x
2
x
6
x
log 6
x
log
×
=
×
=
=
=
6
6
8
8
8
3
3
log 6
x
=
log
x
log 6
=
log
6
6
8
6
6
8
3
log
3
2
8
x =
log 8
=
6
log 6
2
3
log
log 3
−
log 8
a
−
3
2
8
=
2
2
=
log 6
log 3
+
log 2
a
+
1
2
2
2
as 6x and takes logs base 6 of both sides
M1: Divides by 24, writes 3
x
×
2
x
3
3
which may be implied by e.g.
6
x
x
log
=
=
6
8
8
B1: For
log 6
x
=
x
log 6
(may be implied)
6
6
dM1: Changes to log base 2 correctly (and makes x the subject)
A1: Correct expression
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with its registered office at 80 Strand, London, WC2R 0RL, United Kingdom
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