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

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

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 2

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 3

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 4

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 5

SLPaper 1

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

[6]

Question 6

SLPaper 1

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

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

Find the value of $k$ .

[5]

Question 7

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 8

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 and their maximal domains.

[4]

Solve $(x+9)^k=\ln(9+e^{kx})$ for $x\ge0$ (3 d.p.) and justify uniqueness.

[7]

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

[4]

Question 9

SLPaper 2

Let $f(x)=e^{kx},\quad g(x)=\ln(x+7),\quad x>-7,\ k>0.$ It is given that $(f\circ g)(0)=3$ .

Determine $k$ correct to 3 significant figures.

[3]

Find the composites and state maximal domains.

[4]

Solve $(x+7)^k=\ln(7+e^{kx})$ for $x\ge0$ (3 d.p.) and justify uniqueness.

[7]

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

[4]

Question 10

SLPaper 2

Consider the equation $e^{0.4x}=\sin x+2$ for $x \geq 0$ .

Use technology to solve the equation for $0\le x\le12$ . Give all solutions correct to 3 d.p.

[4]

By considering the graphs, justify why no further solutions exist for $x>12$ .

[3]

Consider instead $e^{0.4x}=\cos x+3$ . State and justify the number of real solutions for $x\ge0$ .

[4]

SL 2.9—Exponential and logarithmic functions Questionbank