SL 2.9—Exponential and logarithmic functions - IB Questionbank | RevisionDojo

SL 2.1—Equations of straight lines, parallel and perpendicular

SL 2.2—Functions, notation domain, range and inverse as reflection

SL 2.3—Graphing

SL 2.4—Key features of graphs, intersections using technology

SL 2.5—Composite functions, identity, finding inverse

SL 2.6—Quadratic function

SL 2.7—Solutions of quadratic equations and inequalities, discriminant and nature of roots

SL 2.8—Reciprocal and simple rational functions, equations of asymptotes

SL 2.9—Exponential and logarithmic functions

SL 2.10—Solving equations graphically and analytically

SL 2.11—Transformation of functions

AHL 2.12—Factor and remainder theorems, sum and product of roots

AHL 2.13—Rational functions

AHL 2.14—Odd and even functions, self-inverse, inverse and domain restriction

AHL 2.15—Solutions of inequalities

AHL 2.16—Graphing modulus equations and inequalities

SL 2.9—Exponential and logarithmic functions Questionbank

Practice SL 2.9—Exponential and logarithmic functions with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

Solve the equation $2^{x+1} \cdot 3^{x-1}=18$ . Give your answer in the form $\frac{\ln a}{\ln b}$ , where $a, b \in \mathbb{N}$ .

Show that the equation can be rewritten as $\ln \left(2^{x+1} \cdot 3^{x-1}\right)=\ln 18$ .

[1]

Find the exact value of $x$ .

[3]

Question 2

SLPaper 1

The function is defined as $f(x)=3^{x}+1$ . The graph of $f(x)$ passes through points $A(0, a)$ , $B(1, b)$ , and $C(-1, c)$ .

Find the values of $a, b$ , and $c$ .

[3]

Sketch the graph of $f(x)$ . Indicate the y-intercept, the horizontal asymptote, and the points $A, B$ , and $C$ on the graph.

[3]

Question 3

SLPaper 1

The function is defined as $f(x)=e^{x-2}-3$ . The graph of $f(x)$ has an x-intercept at point $P$ .

Find the coordinates of point $P$ .

[3]

Sketch the graph of $f(x)$ . Indicate the x-intercept, y-intercept, and horizontal asymptote.

[3]

Question 4

SLPaper 1

Consider the curve with equation $y=(2x-1)e^{kx}$ , where $x \in \mathbb{R}$ and $k \in \mathbb{Q}$ .

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

Find the value of $k$ .

[5]

Question 5

SLPaper 1

The first two terms of an infinite geometric sequence are $v_1 = 18$ and $v_2 = 12\sin^{2} \alpha$ , where $0 < \alpha < 2\pi$ , and $\alpha \neq \pi$ .

Find an expression for $r$ in terms of $\alpha$ .

[2]

Find the values of $\alpha$ which give the greatest value of the sum.

[6]

Question 6

SLPaper 1

Let $f(x)=\log _{3}(x+2)+1$ , for $x>-2$ . The graph of $f$ passes through point $B(1, b)$ .

Find the value of $b$ .

[2]

Solve the equation $f(x)=2$ . Express your answer in exact form.

[3]

Sketch the graph of $f$ . Indicate the vertical asymptote, the x-intercept, and point $B$ .

[3]

Question 7

SLPaper 1

Solve the equation $\log _{2}(x)+\log _{2}(x+4)=3$ .

[6]

Question 8

SLPaper 2

The population of a certain species of fish in a lake is modeled by the function $P(t) = 500 \ln(t + 1)$ , where $P(t)$ is the population at time $t$ in years.

Find the population of the fish after 4 years.

[2]

Determine the time it will take for the population to reach 1000 fish.

[2]

Question 9

SLPaper 2

Let $f(x)=e^{kx},\quad g(x)=\ln(x+9),\quad x>-9,\ k>0.$ It is given that $(f\circ g)(1)=7$ .

Determine $k$ correct to 3 significant figures.

[3]

Find the composites $(f\circ g)(x)$ and $(g\circ f)(x)$ and state their maximal domains.

[4]

Solve $(x+9)^k=\ln(9+e^{kx})$ for $x\ge0$ , giving your answer correct to 3 decimal places, and justify the uniqueness of the solution.

[7]

Find $(f\circ g)^{-1}(y)$ and state its domain and range.

[4]

Question 10

SLPaper 1

Consider the function $f(x) = \log_3(x^2 - 4)$ .

State the domain of $f(x)$ .

[2]

Given that the domain of $f$ is $x > 2$ , find the inverse function $f^{-1}(x)$ .

[4]

Paper

Level

Question 1

SLPaper 1

Solve the equation $2^{x+1} \cdot 3^{x-1}=18$ . Give your answer in the form $\frac{\ln a}{\ln b}$ , where $a, b \in \mathbb{N}$ .

Show that the equation can be rewritten as $\ln \left(2^{x+1} \cdot 3^{x-1}\right)=\ln 18$ .

[1]

Find the exact value of $x$ .

[3]

Question 2

SLPaper 1

The function is defined as $f(x)=3^{x}+1$ . The graph of $f(x)$ passes through points $A(0, a)$ , $B(1, b)$ , and $C(-1, c)$ .

Find the values of $a, b$ , and $c$ .

[3]

Sketch the graph of $f(x)$ . Indicate the y-intercept, the horizontal asymptote, and the points $A, B$ , and $C$ on the graph.

[3]

Question 3

SLPaper 1

The function is defined as $f(x)=e^{x-2}-3$ . The graph of $f(x)$ has an x-intercept at point $P$ .

Find the coordinates of point $P$ .

[3]

Sketch the graph of $f(x)$ . Indicate the x-intercept, y-intercept, and horizontal asymptote.

[3]

Question 4

SLPaper 1

Consider the curve with equation $y=(2x-1)e^{kx}$ , where $x \in \mathbb{R}$ and $k \in \mathbb{Q}$ .

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

Find the value of $k$ .

[5]

Question 5

SLPaper 1

The first two terms of an infinite geometric sequence are $v_1 = 18$ and $v_2 = 12\sin^{2} \alpha$ , where $0 < \alpha < 2\pi$ , and $\alpha \neq \pi$ .

Find an expression for $r$ in terms of $\alpha$ .

[2]

Find the values of $\alpha$ which give the greatest value of the sum.

[6]

Question 6

SLPaper 1

Let $f(x)=\log _{3}(x+2)+1$ , for $x>-2$ . The graph of $f$ passes through point $B(1, b)$ .

Find the value of $b$ .

[2]

Solve the equation $f(x)=2$ . Express your answer in exact form.

[3]

Sketch the graph of $f$ . Indicate the vertical asymptote, the x-intercept, and point $B$ .

[3]

Question 7

SLPaper 1

Solve the equation $\log _{2}(x)+\log _{2}(x+4)=3$ .

[6]

Question 8

SLPaper 2

The population of a certain species of fish in a lake is modeled by the function $P(t) = 500 \ln(t + 1)$ , where $P(t)$ is the population at time $t$ in years.

Find the population of the fish after 4 years.

[2]

Determine the time it will take for the population to reach 1000 fish.

[2]

Question 9

SLPaper 2

Let $f(x)=e^{kx},\quad g(x)=\ln(x+9),\quad x>-9,\ k>0.$ It is given that $(f\circ g)(1)=7$ .

Determine $k$ correct to 3 significant figures.

[3]

Find the composites $(f\circ g)(x)$ and $(g\circ f)(x)$ and state their maximal domains.

[4]

Solve $(x+9)^k=\ln(9+e^{kx})$ for $x\ge0$ , giving your answer correct to 3 decimal places, and justify the uniqueness of the solution.

[7]

Find $(f\circ g)^{-1}(y)$ and state its domain and range.

[4]

Question 10

SLPaper 1

Consider the function $f(x) = \log_3(x^2 - 4)$ .

State the domain of $f(x)$ .

[2]

Given that the domain of $f$ is $x > 2$ , find the inverse function $f^{-1}(x)$ .

[4]