IB Questionbank

User interface language: English | Español

Date	November Example questions	Marks available	5	Reference code	EXN.1.SL.TZ0.2
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Solve	Question number	2	Adapted from	N/A

Question

Solve the equation $2 \ln x = \ln 9 + 4$ . Give your answer in the form $x = p e^{q}$ where $p, q \in ℤ^{+}$ .

Markscheme

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

METHOD 1

$2 \ln x - \ln 9 = 4$

uses $m \ln x = \ln x^{m}$ (M1)

$\ln x^{2} - \ln 9 = 4$

uses $\ln a - \ln b = \ln \frac{a}{b}$ (M1)

$\ln \frac{x^{2}}{9} = 4$

$\frac{x^{2}}{9} = e^{4}$ A1

$x^{2} = 9 e^{4} \Rightarrow x = \sqrt{9 e^{4}} (x > 0)$ A1

$x = 3 e^{2} (p = 3, q = 2)$ A1

METHOD 2

expresses $4$ as $4 \ln e$ and uses $\ln x^{m} = m \ln x$ (M1)

$2 \ln x = 2 \ln 3 + 4 \ln e (\ln x = \ln 3 + 2 \ln e)$ A1

uses $2 \ln e = \ln e^{2}$ and $\ln a + \ln b = \ln a b$ (M1)

$\ln x = \ln (3 e^{2})$ A1

$x = 3 e^{2} (p = 3, q = 2)$ A1

METHOD 3

expresses $4$ as $4 \ln e$ and uses $m \ln x = \ln x^{m}$ (M1)

$\ln x^{2} = \ln 3^{2} + \ln e^{4}$ A1

uses $\ln a + \ln b = \ln a b$ (M1)

$\ln x^{2} = \ln (3^{2} e^{4})$

$x^{2} = 3^{2} e^{4} \Rightarrow x = \sqrt{3^{2} e^{4}} (x > 0)$ A1

so $x = 3 e^{2} (x > 0) (p = 3, q = 2)$ A1

[5 marks]

Examiners report

[N/A]

Syllabus sections

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

Show 49 related questions

22M.1.SL.TZ1.8b.i:
Show that $p = \frac{2}{3}$ .
22M.1.AHL.TZ1.10b.ii:
Write down $d$ in the form $k \ln x$ , where $k \in ℚ$ .
22M.1.SL.TZ2.8b:
Find the value of $k$ and the value of $p$ .
22M.2.AHL.TZ2.12e:
By solving the logistic differential equation, show that its solution can be expressed in the form

$k t = \ln \frac{P}{P_{0}} (\frac{N - P_{0}}{N - P})$ .
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.
SPM.1.SL.TZ0.6b:
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 ${\left( {1251} \right)_n} + {\left( {30} \right)_n} = {\left( {504} \right)_n} + {\left( {504} \right)_n}$ .

Find the value of $n$ .
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:
The sum of the first $n$ terms of the series is $\ln (\frac{1}{x^{3}})$ .

Find the value of $n$ .
18M.1.AHL.TZ1.H_5:
Solve ${\left( {{\text{ln}}\,x} \right)^2} - \left( {{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x} \right) < 2{\left( {{\text{ln}}\,2} \right)^2}$ .
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:
Write down the remainder when $14^{2022}$ is divided by $7$ .
19N.3.AHL.TZ0.Hdm_2b:
Convert ${\left( {7A2} \right)_{16}}$ to base $5$ , where ${\left( A \right)_{16}} = {\left( {10} \right)_{10}}$ .
19N.3.AHL.TZ0.Hdm_2a.ii:
Find the remainder when ${15^{1207}}$ is divided by $13$ .
18M.2.SL.TZ1.S_8b:
Use the regression equation to estimate the value of y when x = 3.57.
19M.1.AHL.TZ2.H_7:
Solve the simultaneous equations

${\text{lo}}{{\text{g}}_2}6x = 1 + 2\,{\text{lo}}{{\text{g}}_2}y$

$1 + {\text{lo}}{{\text{g}}_6}x = {\text{lo}}{{\text{g}}_6}\left( {15y - 25} \right)$ .
20N.3.AHL.TZ0.Hdm_3b.ii:
The base $13$ number $1 y 93 y 25$ is divisible by $6$ . Find the possible values of the digit $y$ .
18M.1.AHL.TZ2.H_11a:
Show that ${\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{2}{\text{lo}}{{\text{g}}_r}\,x$ where $r,\,x \in {\mathbb{R}^ + }$ .
20N.3.AHL.TZ0.Hdm_3a.ii:
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:
Prove that a number in base $13$ is divisible by $6$ if, and only if, the sum of its digits is divisible by $6$ .
18M.2.SL.TZ1.S_8a:
Find the value of a and of b.
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.
22M.1.AHL.TZ1.10a.i:
Show that $p = \pm \frac{1}{\sqrt{3}}$ .
22M.1.SL.TZ1.2a:
The expression $\frac{3 \sqrt{x} - 5}{\sqrt{x}}$ can be written as $3 - 5 x^{p}$ . Write down the value of $p$ .
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:
The relationship between x and y can be modelled using the formula y = kxⁿ, where k ≠ 0 , n ≠ 0 , n ≠ 1.

By expressing ln y in terms of ln x, find the value of n and of k.
19N.3.AHL.TZ0.Hdm_2a.i:
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:
Hence find the exact value of $\int_{0}^{3} \frac{1}{f (x)} d x$ , expressing your answer as a single logarithm.
21N.1.AHL.TZ0.3:
Solve the equation $\log_{3} \sqrt{x} = \frac{1}{2 \log_{2} 3} + \log_{3} (4 x^{3})$ , where $x > 0$ .
21N.1.SL.TZ0.8a:
Show that $a = 8$ .
21N.1.SL.TZ0.8d.ii:
Find the value of $p$ and the value of $q$ .
21N.1.SL.TZ0.8d.i:
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:
Write down the equation of the vertical asymptote of the graph of $f$ .

Hide related questions

Topic 1—Number and algebra

Question

Markscheme

Examiners report

Syllabus sections

View options