Exponentials, Logarithms, e and ln

Read a log as a power, use e and ln as inverses, and sketch exponential and log curves with exact intercepts and asymptotes.

In the exam for Edexcel Further Pure Maths 4PM1, IGCSE Additional Maths 0606 · 23 real past-paper questions below

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Growth by multiplying

A video goes viral: one share becomes 2, then 4, then 8. Each step multiplies by the same number. When the step number sits in the power, you have an exponential function.

Formula

y = a^{x}

$a$ — the base: the number you multiply by each step. It is positive and not 1.
$x$ — the power. It can be any number: whole, negative or a fraction.

Worked example

1
Find the value of $y = 2^{x}$ when $x = - 2$ , $x = 0$ and $x = 3$ .
2
$2^{- 2} = \frac{1}{4}$ — A negative power means one over: $\frac{1}{2 ^{2}}$ .
3
$2^{0} = 1$ — Any base to the power 0 is 1.
4
$2^{3} = 8$ — Three 2s multiplied: 2 × 2 × 2.

Now let x get very negative: $2^{- 10} = \frac{1}{1024}$ . The curve gets closer and closer to the x-axis, but it never touches it.

Rule

A line that a curve gets closer and closer to, but never reaches, is called an asymptote. For $y = a^{x}$ the asymptote is the x-axis, $y = 0$ .

It crosses the y-axis at (0, 1), because $a^{0} = 1$ .
y is always positive. The curve stays above the x-axis.
If $a > 1$ the curve rises to the right (growth). If $0 < a < 1$ , such as $(\frac{1}{2})^{x} = 2^{- x}$ , it falls to the right (decay).

Common mistake

Drawing the curve touching or crossing the x-axis. $a^{x}$ is never 0 and never negative.

Key idea

Every curve $y = a^{x}$ passes through (0, 1), stays above the x-axis, and has the asymptote $y = 0$ .

What a log asks

Now turn the question round. You know the base, 2, and the answer, 8. Which power gives 8? The tool that finds a power is called a logarithm, or log for short.

Formula

lo g_{a} x = y ⟺ a^{y} = x

$a$ — the base. It is positive and not 1.
$x$ — the number you want to reach. It must be positive.
$y$ — the power. This is the answer the log gives.

Read $lo g_{2} 8$ as “log base 2 of 8”. It asks: 2 to what power gives 8? The answer is 3.

Worked example

1
Find the value of each log without a calculator.
2
$lo g_{3} 81 = 4$ — $3^{4} = 81$ .
3
$lo g_{5} \frac{1}{25} = - 2$ — $5^{- 2} = \frac{1}{25}$ . Numbers below 1 have negative logs.
4
$lo g_{9} 3 = \frac{1}{2}$ — $9^{\frac{1}{2}} = 9 = 3$ .
5
$l g 0.001 = - 3$ — lg means base 10, and $1 0^{- 3} = 0.001$ .

Two bases have their own short names. $l g x$ means $lo g_{10} x$ , and $ln x$ means $lo g_{e} x$ . You meet e in step 4.

Worked example

1
Write $6^{2} = 36$ in log form. Then write $lo g_{4} x = 3$ in index form and find x.
2
$lo g_{6} 36 = 2$ — The base stays the base. The power becomes the answer of the log.
3
$x = 4^{3}$ — The log's answer, 3, becomes the power on the base 4.
4
$x = 64$ — 4 × 4 × 4 = 64.

Rule

You can only take the log of a positive number. No power of a positive base gives 0 or a negative number, so $lo g_{2} 0$ and $lo g_{2} (- 4)$ do not exist.

Common mistake

Dividing: $lo g_{2} 8$ is not $8 \div 2 = 4$ . Ask which power: $2^{3} = 8$ , so the answer is 3.

Key idea

A log is a power. Whenever you see a log, ask: the base to what power gives this number?

The mirror graph

$y = 2^{x}$ takes a power and gives a number. $y = lo g_{2} x$ takes the number and gives back the power. Each one undoes the other, so they are inverse functions.

Rule

The graph of an inverse function is the reflection of the first graph in the line $y = x$ . Every point (p, q) on one graph becomes (q, p) on the other.

Worked example

1
Use points on $y = 2^{x}$ to find points on $y = lo g_{2} x$ .
2
$(0, 1) \to (1, 0)$ — $2^{0} = 1$ , so $lo g_{2} 1 = 0$ . Swap the coordinates.
3
$(1, 2) \to (2, 1)$ — $2^{1} = 2$ , so $lo g_{2} 2 = 1$ .
4
$(3, 8) \to (8, 3)$ — $2^{3} = 8$ , so $lo g_{2} 8 = 3$ .
5
$(- 1, \frac{1}{2}) \to (\frac{1}{2}, - 1)$ — Between 0 and 1, the log is negative.

	$y = a^{x}$	$y = lo g_{a} x$
Crosses an axis at	(0, 1)	(1, 0)
Asymptote	$y = 0$	$x = 0$
x can be	any number	positive only
y can be	positive only	any number

For a base above 1, the log curve keeps rising, but more and more slowly. All boards expect these shapes. 4PM1 uses whole-number bases above 1, such as 2, 3 or 10, and 0606 and 4037 allow any base.

Common mistake

Drawing the log curve to the left of the y-axis. You cannot take the log of 0 or a negative number, so the curve lives only where $x > 0$ .

Key idea

Exponential and log graphs are mirror images in $y = x$ . The asymptote $y = 0$ becomes $x = 0$ , and (0, 1) becomes (1, 0).

The number e and ln

One base is special: the number e. The curve $y = e^{x}$ has a gradient equal to its own height at every point. That makes it the natural base for growth and decay in science.

Formula

e \approx 2.718

$e$ — a fixed number, like π. Its decimals never end and never repeat. Your calculator has an $e^{x}$ key.

Formula

l n x = lo g_{e} x

$ln$ — the natural logarithm: a log with base e. Your calculator has an ln key.

$y = e^{x}$ and $y = ln x$ have the shapes you already know. $y = e^{x}$ passes through (0, 1) with asymptote $y = 0$ . $y = ln x$ passes through (1, 0) with asymptote $x = 0$ .

They are inverse functions, so each one undoes the other. Learn these two facts. No board prints them for you.

Formula

e^{l n x} = x

$x$ — any positive number. Take ln, then raise e to it, and you are back where you started.

Formula

ln (e^{x}) = x

$x$ — any number at all. Raise e to it, then take ln, and you are back to x.

Worked example

1
Simplify without a calculator.
2
$ln (e^{5}) = 5$ — ln undoes e to a power.
3
$e^{l n 7} = 7$ — e to a power undoes ln.
4
$ln 1 = 0$ — $e^{0} = 1$ .
5
$ln e = 1$ — $e^{1} = e$ .

Worked example

1
Find the exact value of x when $ln x = 3$ , and when $e^{x} = 10$ .
2
$ln x = 3 \Rightarrow x = e^{3}$ — Raise e to both sides. e undoes ln.
3
$e^{x} = 10 \Rightarrow x = ln 10$ — Take ln of both sides. ln undoes e.

Rule

No calculator on 0606 and 4037 Paper 1. Leave answers like $e^{3}$ and $ln 10$ exactly as they are.

Key idea

In 1683 Jacob Bernoulli asked what happens to a loan if the interest is added more and more often. The answer settles on about 2.718. Leonhard Euler later gave this number the letter e.

Key idea

e and ln undo each other, just like squaring and square-rooting a positive number.

Sketching exponential curves

0606 and 4037 questions ask you to sketch curves like $y = 3 e^{x} - 6$ . 4PM1 asks for the same intercepts and asymptotes inside longer curve questions. A full sketch needs three things: the right shape, the points where it crosses the axes, and the asymptote.

Formula

y = k e^{n x} + a

$k$ — stretches the curve. If k is negative, the curve flips upside down.
$n$ — sets the direction. With k positive, a positive n rises to the right and a negative n falls to the right.
$a$ — moves the curve up or down. The asymptote is $y = a$ .

Why is the asymptote $y = a$ ? On one side, $e^{n x}$ gets closer and closer to 0. So y gets closer and closer to a, but never reaches it.

Steps

1
1. y-intercept: put $x = 0$ . Since $e^{0} = 1$ , you get $y = k + a$ .
2. Asymptote: write $y = a$ .
3. x-intercept: put $y = 0$ and solve, if it has a solution.
4. Draw the shape through the intercepts, getting closer to the asymptote.

Worked example

1
Sketch $y = 3 e^{x} - 6$ . State the intercepts with the axes and the asymptote.
2
$y = 3 e^{0} - 6 = - 3$ — Put x = 0. The y-intercept is (0, −3).
3
$y = - 6$ — The asymptote. As x gets very negative, $e^{x}$ gets close to 0.
4
$3 e^{x} - 6 = 0 \Rightarrow e^{x} = 2$ — Put y = 0 for the x-intercept.
5
$x = ln 2$ — Take ln. The x-intercept is (ln 2, 0).
6
$k = 3, n = 1$ — Both positive: the curve rises to the right and flattens onto $y = - 6$ on the left.

Worked example

1
Sketch $y = 4 e^{- 2 x} + 1$ .
2
$y = 4 + 1 = 5$ — Put x = 0. The y-intercept is (0, 5).
3
$y = 1$ — The asymptote.
4
$4 e^{- 2 x} + 1 > 1$ — $4 e^{- 2 x}$ is always positive, so y is never 0. No x-intercept.
5
$n = - 2$ — Negative n: the curve falls to the right and flattens onto $y = 1$ .

Rule

Mark schemes give separate marks for the shape, each intercept and the asymptote. Write the asymptote as an equation, such as $y = - 6$ , and label the intercepts on your sketch.

Common mistake

Writing $y = 0$ as the asymptote out of habit. The $+ a$ moves the asymptote to $y = a$ .

Key idea

Put x = 0, read the asymptote from the plain number, then put y = 0.

Sketching log curves

Log curves follow the same plan, but the asymptote is vertical. You can only take the log of a positive number, so the curve stops where the inside becomes 0.

Formula

y = k ln (a x + b)

$k$ — stretches the curve. If k is negative, the curve flips upside down.
$a x + b$ — the inside of the log. The curve exists only where it is positive.

Steps

1
1. Asymptote: solve inside = 0.
2. x-intercept: solve inside = 1, because $ln 1 = 0$ .
3. y-intercept: put $x = 0$ , giving $y = k ln b$ . This exists only if b is positive.
4. Draw the curve coming up from the asymptote and through the intercepts.

Worked example

1
Sketch $y = 2 ln (2 x + 5)$ . State the intercepts and the asymptote.
2
$2 x + 5 = 0 \Rightarrow x = - \frac{5}{2}$ — Inside = 0. The asymptote is $x = - \frac{5}{2}$ .
3
$2 x + 5 = 1 \Rightarrow x = - 2$ — Inside = 1. The x-intercept is (−2, 0).
4
$y = 2 ln 5$ — Put x = 0. The y-intercept is (0, 2 ln 5). Leave it exact.
5
$k = 2, a = 2$ — Both positive: the curve rises to the right and plunges down beside $x = - \frac{5}{2}$ .

Worked example

1
Sketch $y = lo g_{3} (x + 9)$ .
2
$x + 9 = 0 \Rightarrow x = - 9$ — The asymptote is $x = - 9$ .
3
$x + 9 = 1 \Rightarrow x = - 8$ — The log of 1 is 0 in every base. The x-intercept is (−8, 0).
4
$y = lo g_{3} 9 = 2$ — Put x = 0. $3^{2} = 9$ , so the y-intercept is (0, 2).

Rule

If a is negative, as in $y = ln (4 - x)$ , the curve is flipped left to right. The asymptote is on the right, here $x = 4$ , and the curve falls as x increases.

Common mistake

Adding a horizontal asymptote to a log curve. It keeps rising slowly for ever. Its only asymptote is the vertical one, and an extra wrong asymptote can cost the mark.

Key idea

Inside = 0 gives the asymptote. Inside = 1 gives the x-intercept. x = 0 gives the y-intercept.

Which method? Read the question

Before you start, read what the question asks. The wording tells you which idea to use.

You see	You do
“Find $lo g_{a} x$ ” exactly	Ask: a to what power gives x?
$e^{l n x}$ or $ln (e^{x})$	They cancel. The answer is x
“Sketch $y = k e^{n x} + a$ ”	x = 0, asymptote $y = a$ , then y = 0
“Sketch $y = k ln (a x + b)$ ”	Inside = 0, inside = 1, then x = 0
The inverse of $e^{x}$ or $ln x$	Reflect in $y = x$ : swap each point
“State any asymptote”	Write an equation: $y = \dots$ or $x = \dots$

Common mistake

Turning exact intercepts like $2 ln 5$ into decimals on the non-calculator paper. Leave them exact.

Key idea

Exponential curves flatten onto a horizontal line. Log curves rise from a vertical line. Each is the mirror image of the other in $y = x$ .

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