Sec 3 A-Math · Exponential and logarithmic functions

Graphs of exponential and logarithmic functions questions with answers

Practice questions on graphs of exponential and logarithmic functions, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

The diagram shows four sketches, $A$ , $B$ , $C$ and $D$ , with their asymptotes. Which one is the graph of $y = 3 - 2e^{x}$ ?

Four sketches A to D of exponential or logarithmic curves, each with its asymptote dashed and labelled.

Step 1 of 3

The asymptote: e to a power is never 0, but it gets as close to 0 as you like: then y gets close to the number term

x \to -\infty

e^{x} \to 0

, so

y \to 3

y = 3

Try a question: Graphs of exponential and logarithmic functions

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Questions

Question 1Choose the sketch
The diagram shows four sketches, $A$ , $B$ , $C$ and $D$ , with their asymptotes. Which one is the graph of $y = e^{x}$ ?
Show the worked solution
1. Step 1: The asymptote: e to a power is never 0, but it gets as close to 0 as you like: then y gets close to the number termAs $x \to -\infty$ , $e^{x} \to 0$ , so $y \to 0$ . $y = 0$
2. Step 2: As x increases, does the curve go up or down? $y$ increases as $x$ increases ( $1e^{x}$ goes up).
3. Step 3: Match the asymptote and the direction to a sketchSketch $B$
Question 2The asymptote
State the equation of the asymptote of the curve $y = 3 + 2e^{-x}$ .
Show the worked solution
1. Step 1: e to a power is never 0, but it gets as close to 0 as you like: then y gets close to the number termAs $x \to \infty$ , $e^{-x} \to 0$ , so $y \to 3$ . $y = 3$
Question 3Where the curve meets an axis
Find the exact $x$ -coordinate of the point where the curve $y = 2 - \ln(x - 1)$ meets the $x$ -axis.
Show the worked solution
1. Step 1: On the x-axis, y = 0: make the ln the subject $2 - \ln(x - 1) = 0$ $\ln(x - 1) = 2$
2. Step 2: ln(something) = k means something = eᵏ $x - 1 = e^{2}$ $x = e^{2} + 1$
Question 4Number of solutions: draw a line
The diagram shows the curve $y = -\ln x$ and its asymptote. By drawing a suitable straight line on the diagram, state the number of solutions of the equation $-\ln x + 2x = 3$ .
Show the worked solution
1. Step 1: Rearrange so that the left side is the curve's own equation: the right side is then the line to draw $-\ln x = -2x + 3$ Draw the line $y = -2x + 3$ .
2. Step 2: Draw that line on the diagram and count the points where it meets the curveThe line $y = -2x + 3$ meets the curve at 2 points.

More on exponential and logarithmic functions

Notes: Exponential and logarithmic functions Worksheet with answers (PDF)

Endless questions on graphs of exponential and logarithmic functions

A free account gives you new questions every time, step-by-step help, and tracks what you've mastered.

Start free

Worked example

The diagram shows four sketches,

A

B

C

and

D

, with their asymptotes. Which one is the graph of

y = 3 - 2e^{x}

Step 1 of 3

The asymptote: e to a power is never 0, but it gets as close to 0 as you like: then y gets close to the number term

x \to -\infty

e^{x} \to 0

, so

y \to 3

y = 3

Questions

Question 1Choose the sketch

The diagram shows four sketches, $A$ , $B$ , $C$ and $D$ , with their asymptotes. Which one is the graph of $y = e^{x}$ ?

Show the worked solution

Step 1: The asymptote: e to a power is never 0, but it gets as close to 0 as you like: then y gets close to the number termAs $x \to -\infty$ , $e^{x} \to 0$ , so $y \to 0$ . $y = 0$
Step 2: As x increases, does the curve go up or down? $y$ increases as $x$ increases ( $1e^{x}$ goes up).
Step 3: Match the asymptote and the direction to a sketchSketch $B$

Question 2The asymptote

State the equation of the asymptote of the curve $y = 3 + 2e^{-x}$ .

Show the worked solution

Step 1: e to a power is never 0, but it gets as close to 0 as you like: then y gets close to the number termAs $x \to \infty$ , $e^{-x} \to 0$ , so $y \to 3$ . $y = 3$

Question 3Where the curve meets an axis

Find the exact $x$ -coordinate of the point where the curve $y = 2 - \ln(x - 1)$ meets the $x$ -axis.

Show the worked solution

Step 1: On the x-axis, y = 0: make the ln the subject $2 - \ln(x - 1) = 0$ $\ln(x - 1) = 2$
Step 2: ln(something) = k means something = eᵏ $x - 1 = e^{2}$ $x = e^{2} + 1$

Question 4Number of solutions: draw a line

The diagram shows the curve $y = -\ln x$ and its asymptote. By drawing a suitable straight line on the diagram, state the number of solutions of the equation $-\ln x + 2x = 3$ .

The curve y = -ln x with its asymptote dashed.

Show the worked solution

Step 1: Rearrange so that the left side is the curve's own equation: the right side is then the line to draw $-\ln x = -2x + 3$ Draw the line $y = -2x + 3$ .
Step 2: Draw that line on the diagram and count the points where it meets the curveThe line $y = -2x + 3$ meets the curve at 2 points.