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Date	November 2021	Marks available	5	Reference code	21N.1.AHL.TZ0.3
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Solve	Question number	3	Adapted from	N/A

Question

Solve the equation $\log_{3} \sqrt{x} = \frac{1}{2 \log_{2} 3} + \log_{3} (4 x^{3})$ , where $x > 0$ .

Markscheme

attempt to use change the base (M1)

$\log_{3} \sqrt{x} = \frac{\log_{3} 2}{2} + \log_{3} (4 x^{3})$

attempt to use the power rule (M1)

$\log_{3} \sqrt{x} = \log_{3} \sqrt{2} + \log_{3} (4 x^{3})$

attempt to use product or quotient rule for logs, $\ln a + \ln b = \ln a b$ (M1)

$\log_{3} \sqrt{x} = \log_{3} (4 \sqrt{2} x^{3})$

Note: The M marks are for attempting to use the relevant log rule and may be applied in any order and at any time during the attempt seen.

$\sqrt{x} = 4 \sqrt{2} x^{3}$

$x = 32 x^{6}$

$x^{5} = \frac{1}{32}$ (A1)

$x = \frac{1}{2}$ A1

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » SL 1.7—Laws of exponents and logs

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22M.2.AHL.TZ2.12e:
By solving the logistic differential equation, show that its solution can be expressed in the form

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21N.2.AHL.TZ0.10d:
Sketch the graph of $f$ for $- 30 \leq x \leq 30$ , clearly indicating the points of intersection with each axis and any asymptotes.
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Hence or otherwise solve ${\text{lo}}{{\text{g}}_3}\left( {{\text{2}}\,{\text{sin}}\,x} \right) = {\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right)$ for $0 < x < \frac{\pi }{2}$ .
19N.3.AHL.TZ0.Hdm_2c:
Consider the equation ${\left( {1251} \right)_n} + {\left( {30} \right)_n} = {\left( {504} \right)_n} + {\left( {504} \right)_n}$ .

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SPM.1.SL.TZ0.6a:
Show that ${\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right) = {\text{lo}}{{\text{g}}_3}\sqrt {{\text{cos}}\,2x + 2}$ .
22M.1.SL.TZ1.8a.ii:
Given that $p > 0$ and $S_{\infty} = 3 + \sqrt{3}$ , find the value of $x$ .
22M.1.SL.TZ1.8b.ii:
Write down $d$ in the form $k \ln x$ , where $k \in ℚ$ .
22M.1.AHL.TZ1.10a.ii:
Hence or otherwise, show that the series is convergent.
22M.1.AHL.TZ1.10a.iii:
Given that $p > 0$ and $S_{\infty} = 3 + \sqrt{3}$ , find the value of $x$ .
22M.1.AHL.TZ1.10b.i:
Show that $p = \frac{2}{3}$ .
22M.1.AHL.TZ1.10b.iii:
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Find the value of $n$ .
18M.1.AHL.TZ1.H_5:
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17N.1.AHL.TZ0.H_1:
Solve the equation ${\log _2}(x + 3) + {\log _2}(x - 3) = 4$ .
20N.3.AHL.TZ0.Hdm_3a.i:
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19N.3.AHL.TZ0.Hdm_2b:
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19N.3.AHL.TZ0.Hdm_2a.ii:
Find the remainder when ${15^{1207}}$ is divided by $13$ .
18M.2.SL.TZ1.S_8b:
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19M.1.AHL.TZ2.H_7:
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20N.3.AHL.TZ0.Hdm_3b.ii:
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Show that ${\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{2}{\text{lo}}{{\text{g}}_r}\,x$ where $r,\,x \in {\mathbb{R}^ + }$ .
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Use Fermat’s little theorem to find the remainder when $14^{2022}$ is divided by $17$ .
17M.2.AHL.TZ2.H_6:
Given that ${\log _{10}}\left( {\frac{1}{{2\sqrt 2 }}\left( {p + 2q} \right)} \right) = \frac{1}{2}\left( {{{\log }_{10}}p + {{\log }_{10}}q} \right),{\text{ }}p > 0,{\text{ }}q > 0$ , find $p$ in terms of $q$ .
20N.3.AHL.TZ0.Hdm_3b.i:
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18M.2.SL.TZ1.S_8a:
Find the value of a and of b.
EXN.1.SL.TZ0.2:
Solve the equation $2 \ln x = \ln 9 + 4$ . Give your answer in the form $x = p e^{q}$ where $p, q \in ℤ^{+}$ .
22M.1.AHL.TZ1.1:
Find the value of $\int_{1}^{9} (\frac{3 \sqrt{x} - 5}{\sqrt{x}}) d x$ .
21N.2.AHL.TZ0.10a.ii:
$y$ -axis.
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Show that $p = \pm \frac{1}{\sqrt{3}}$ .
22M.1.SL.TZ1.2a:
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22M.1.SL.TZ1.5:
Consider the curve with equation $y = (2 x - 1) e^{k x}$ , where $x \in ℝ$ and $k \in ℚ$ .

The tangent to the curve at the point where $x = 1$ is parallel to the line $y = 5 e^{k} x$ .

Find the value of $k$ .
22M.1.SL.TZ1.8a.i:
Show that $p = \pm \frac{1}{\sqrt{3}}$ .
21N.1.SL.TZ0.8b:
Write down an expression for $f^{- 1} (x)$ .
21N.2.AHL.TZ0.10c:
The oblique asymptote of the graph of $f$ can be written as $y = a x + b$ where $a, b \in ℚ$ .

Find the value of $a$ and the value of $b$ .
21N.1.SL.TZ0.8c:
Find the value of $f^{- 1} (\sqrt{32})$ .
17M.1.AHL.TZ1.H_1:
Find the solution of ${\log _2}x - {\log _2}5 = 2 + {\log _2}3$ .
16N.1.AHL.TZ0.H_7:
Solve the equation ${4^x} + {2^{x + 2}} = 3$ .
18M.2.SL.TZ1.S_8c:
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By expressing ln y in terms of ln x, find the value of n and of k.
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State Fermat’s little theorem.
19N.1.AHL.TZ0.H_10d:
The area enclosed by the graph of $y = f\left( x \right)$ and the line $y = 4$ can be expressed as ${\text{ln}}\,v$ . Find the value of $v$ .
21N.2.AHL.TZ0.10e.i:
Express $\frac{1}{f (x)}$ in partial fractions.
21N.2.AHL.TZ0.10e.ii:
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21N.1.SL.TZ0.8a:
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Show that $27, p, q$ and $125$ are four consecutive terms in a geometric sequence.
21N.2.AHL.TZ0.10a.i:
$x$ -axis.
21N.2.AHL.TZ0.10b:
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