LFM Pure and Mechanics

csat-suneung 2026 Q6 3 marks Simplify or Evaluate a Logarithmic Expression View

Two real numbers $a , b$ greater than 1 satisfy $$\log _ { a } b = 3 , \quad \log _ { 3 } \frac { b } { a } = \frac { 1 } { 2 }$$ What is the value of $\log _ { 9 } a b$? [3 points]
(1) $\frac { 3 } { 8 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 5 } { 8 }$
(4) $\frac { 3 } { 4 }$
(5) $\frac { 7 } { 8 }$

gaokao 2015 Q3 5 marks Solve a Logarithmic Inequality View

The domain of the function $f ( \mathrm { x } ) = \log _ { 2 } \left( \mathrm { x } ^ { 2 } + 2 \mathrm { x } - 3 \right)$ is
(A) $[ - 3,1 ]$
(B) $( - 3,1 )$
(C) $( - \infty , - 3 ] \cup [ 1 , + \infty )$
(D) $( - \infty , - 3 ) \cup ( 1 , + \infty )$

gaokao 2015 Q8 Compare or Order Logarithmic Values View

8. Let $a$ and $b$ be positive numbers not equal to $1$. Then ``$3 ^ { a } > 3 ^ { b } > 3$'' is ``$\log _ { a } 3 < \log _ { b } 3$'' a
(A) necessary and sufficient condition
(B) sufficient but not necessary condition
(C) necessary but not sufficient condition
(D) neither sufficient nor necessary condition

gaokao 2015 Q8 Verify Truth of Logarithmic Statements View

8. Let real numbers $a , b , t$ satisfy $| a + 1 | = | \sin b | = t$
[Figure]
(Figure for Question 7)
A. If $t$ is determined, then $b ^ { 2 }$ is uniquely determined
B. If $t$ is determined, then $a ^ { 2 } + 2 a$ is uniquely determined
C. If $t$ is determined, then $\sin \frac { b } { 2 }$ is uniquely determined
D. If $t$ is determined, then $a ^ { 2 } + a$ is uniquely determined

II. Fill-in-the-Blank Questions (This section contains 7 questions. Multi-blank questions are worth 6 points each, single-blank questions are worth 4 points each, 36 points total.)

gaokao 2015 Q9 Simplify or Evaluate a Logarithmic Expression View

9. Calculate: $\log _ { 2 } \frac { \sqrt { 2 } } { 2 } =$ $\_\_\_\_$ , $2 ^ { \log _ { 2 } 3 + \log _ { 4 } 3 } =$ $\_\_\_\_$.

gaokao 2015 Q12 Simplify or Evaluate a Logarithmic Expression View

12. The value of $\lg 0.01 + \log _ { 2 } 16$ is \_\_\_\_.

gaokao 2015 Q12 Optimize a Logarithmic Expression View

12. Given $a > 0 , b > 0 , ab = 8$, then when $a$ equals $\_\_\_\_$, $\log _ { 2 } a \cdot \log _ { 2 } ( 2 b )$ attains its maximum value.

gaokao 2015 Q12 Simplify or Evaluate a Logarithmic Expression View

12. If $a = \log _ { 2 } 3$ , then $2 ^ { a } + 2 ^ { - a } =$ $\_\_\_\_$ .

gaokao 2017 Q8 Analyze a Logarithmic Function (Limits, Monotonicity, Zeros, Extrema) View

The monotone increasing interval of the function $f(x) = \ln(x^2 - 2x - 9)$ is
A. $(-\infty, -2)$
B. $(-\infty, 1)$
C. $(1, +\infty)$
D. $(4, +\infty)$

gaokao 2017 Q9 5 marks Analyze a Logarithmic Function (Limits, Monotonicity, Zeros, Extrema) View

Given the function $f(x) = \ln x + \ln(2 - x)$, then
A. $f(x)$ is monotonically increasing on $(0, 2)$
B. $f(x)$ is monotonically decreasing on $(0, 2)$
C. $f(x)$ is increasing on $(0, 1)$ and decreasing on $(1, 2)$
D. $f(x)$ is decreasing on $(0, 1)$ and increasing on $(1, 2)$

gaokao 2018 Q12 5 marks Compare or Order Logarithmic Values View

Let $a = \log _ { 0.2 } 0.3, b = \log _ { 2 } 0.3$, then
A. $a + b < ab < 0$
B. $ab < a + b < 0$
C. $a + b < 0 < ab$
D. $ab < 0 < a + b$

gaokao 2018 Q13 5 marks Determine Parameters of a Logarithmic Function View

Given the function $f ( x ) = \log _ { 2 } \left( x ^ { 2 } + a \right)$. If $f ( 3 ) = 1$, then $a = $ \_\_\_\_

gaokao 2019 Q6 5 marks Logarithmic Formula Application (Modeling) View

In astronomy, the brightness of a celestial body can be described by magnitude or luminosity. The magnitude and luminosity of two stars satisfy $m _ { 2 } - m _ { 1 } = \frac { 5 } { 2 } \lg \frac { E _ { 1 } } { E _ { 2 } }$, where the luminosity of a star with magnitude $m _ { k }$ is $E _ { k } ( k = 1,2 )$. Given that the magnitude of the Sun is $- 26.7$ and the magnitude of Sirius is $- 1.45$, the ratio of the luminosity of the Sun to that of Sirius is (A) $10 ^ { 10.1 }$ (B) 10.1 (C) $\lg 10.1$ (D) $10 ^ { - 10.1 }$

gaokao 2019 Q11 5 marks Compare or Order Logarithmic Values View

Let $f ( x )$ be an even function with domain $\mathbf { R }$ that is monotonically decreasing on $( 0 , + \infty )$. Then
A. $f \left( \log _ { 3 } \frac { 1 } { 4 } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right)$
B. $f \left( \log _ { 3 } \frac { 1 } { 4 } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right)$
C. $f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( \log _ { 3 } \frac { 1 } { 4 } \right)$
D. $f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( \log _ { 3 } \frac { 1 } { 4 } \right)$

gaokao 2019 Q14 Solve a Logarithmic Equation View

14. Given that $f ( x )$ is an odd function, and when $x < 0$, $f ( x ) = - \mathrm { e } ^ { a x }$. If $f ( \ln 2 ) = 8$, then $a =$ $\_\_\_\_$ .

gaokao 2020 Q8 5 marks Express One Logarithm in Terms of Another View

If $a \log _ { 3 } 4 = 2$ , then $4 ^ { - a } =$
A. $\frac { 1 } { 16 }$
B. $\frac { 1 } { 9 }$
C. $\frac { 1 } { 8 }$
D. $\frac { 1 } { 6 }$

gaokao 2020 Q10 5 marks Compare or Order Logarithmic Values View

Let $a = \log _ { 3 } 2 , b = \log _ { 5 } 3 , c = \frac { 2 } { 3 }$. Then
A. $a < c < b$
B. $a < b < c$
C. $b < c < a$
D. $c < a < b$

gaokao 2020 Q12 5 marks Compare or Order Logarithmic Values View

Given $5 ^ { 5 } < 8 ^ { 4 } , 13 ^ { 4 } < 8 ^ { 5 }$ . Let $a = \log _ { 5 } 3 , b = \log _ { 8 } 5 , c = \log _ { 13 } 8$ , then
A. $a < b < c$
B. $b < a < c$
C. $b < c < a$
D. $c < a < b$

gaokao 2021 Q4 10 marks Logarithmic Formula Application (Modeling) View

4. Adolescent vision is a matter of widespread social concern. Vision can be measured using a vision chart. Vision data is usually recorded using the five-point recording method and the decimal recording method. The data $l$ in the five-point recording method and the data $v$ in the decimal recording method satisfy $l = 5 + \log_v$. A student's vision data in the five-point recording method is 4.4. Then their vision data in the decimal recording method is approximately ($\sqrt[10]{10} \approx 1.259$)
A. 1.5
B. 1.2
C. 0.8
D. 0.6

gaokao 2021 Q6 Logarithmic Formula Application (Modeling) View

6. Vision of adolescents is a matter of widespread social concern. Vision can be measured using a vision chart. Vision data is usually recorded using the five-point recording method and the decimal recording method. The data $L$ in the five-point recording method and the data $V$ in the decimal recording method satisfy $L = 5 + \lg V$. It is known that a student's vision data in the five-point recording method is 4.9. Then the student's vision data in the decimal recording method is approximately ( $\sqrt [ 10 ] { 10 } \approx 1.259$ )
A. 1.5
B. 1.2
C. 0.8
D. 0.6

gaokao 2021 Q7 Compare or Order Logarithmic Values View

7. Given $a = \log _ { 5 } 2 , b = \log _ { 8 } 3 , c = \frac { 1 } { 2 }$, which of the following judgments is correct? ( )
A. $c < b < a$
B. $b < a < c$
C. $a < c < b$
D. $a < b < c$
【Answer】C 【Solution】【Analysis】Use the monotonicity of logarithmic functions to compare the sizes of $a$, $b$, and $c$, and thus reach the conclusion. 【Detailed Solution】 $a = \log _ { 5 } 2 < \log _ { 5 } \sqrt { 5 } = \frac { 1 } { 2 } = \log _ { 8 } 2 \sqrt { 2 } < \log _ { 8 } 3 = b$, that is, $a < c < b$. Therefore, the answer is: C.

gaokao 2022 Q7 5 marks Compare or Order Logarithmic Values View

Given $a = \log _ { 5 } 2 , b = \log _ { 8 } 3 , c = \frac { 1 } { 2 }$, which of the following judgments is correct?
A. $c < b < a$
B. $b < a < c$
C. $a < c < b$
D. $a < b < c$

gaokao 2022 Q7 Compare or Order Logarithmic Values View

7. Let $a = 0.1 \mathrm { e } ^ { 0.1 }$, $b = \frac { 1 } { 9 }$, $c = - \ln 0.9$. Then
A. $a < b < c$
B. $c < b < a$
C. $c < a < b$
D. $a < c < b$

gaokao 2024 Q7 4 marks Logarithmic Formula Application (Modeling) View

Let the water quality index be $d = \frac { S - 1 } { \ln n }$, and the larger $d$ is, the better the water quality. If $S$ remains constant and $d _ { 1 } = 2.1 , d _ { 2 } = 2.2$, then the relationship between $n _ { 1 }$ and $n_2$ is \_\_\_\_

grandes-ecoles 2024 Q7 Prove a Logarithmic Identity View

Let $g$ be the function defined by
$$\begin{aligned} g : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Deduce that
$$\int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t } \mathrm {~d} t = \frac { \pi } { \sin ( \pi x ) }$$

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LFM Pure and Mechanics

Indices and Surds 107

Exponential Functions 100

Laws of Logarithms 200

II. Fill-in-the-Blank Questions (This section contains 7 questions. Multi-blank questions are worth 6 points each, single-blank questions are worth 4 points each, 36 points total.)

Exponential Equations & Modelling 58

Arithmetic Sequences and Series 271

Geometric Sequences and Series 203

Differentiation from First Principles 37

Tangents, normals and gradients 93

Stationary points and optimisation 384

Applied differentiation 164

Indefinite & Definite Integrals 533

Areas Between Curves 83

Areas by integration 109

Volumes of Revolution 61

Numerical integration 32

Vectors Introduction & 2D 204

Vectors 3D & Lines 233

Travel graphs 11

SUVAT in 2D & Gravity 6

Constant acceleration (SUVAT) 32

Variable acceleration (1D) 40

Variable acceleration (vectors) 26

Forces, equilibrium and resultants 9

Newton's laws and connected particles 18

Pulley systems 5

Motion on a slope 4

Friction 10

Momentum and Collisions 15

Moments 13

Parametric curves and Cartesian conversion 11

Parametric differentiation 15

Parametric integration 4

Projectiles 40