1.4.2 Exponential functions and the number e

From powers to functions

In lesson 1.4.1 the exponent was a fixed number. Now let the exponent be the variable:

f(x) = b^x \qquad (b > 0,\; b \neq 1)

This is an exponential function with base $b$ . In $x^2$ the variable is the base and the exponent is fixed. In $2^x$ the base is fixed and the variable is the exponent.

The base must be positive. With $b = -2$ , the value at $x = \frac{1}{2}$ would be $\sqrt{-2}$ , which has no real value (lesson 1.4.1). A base of 1 is left out because $1^x = 1$ for every $x$ : a flat line.

With a positive base, any real number works as the exponent. So the domain of $b^x$ is all real numbers.

2^x

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Adding against multiplying

A straight line adds the same amount at each step. An exponential multiplies by the same factor at each step. Here are both, starting at 100:

$x$	0	1	2	3	4
$100 + 10x$	100	110	120	130	140
$100 \cdot 1.1^x$	100	110	121	133.1	146.41

The line rises by 10 every time. The exponential rises by 10, then 11, then 12.1, then 13.31, because each rise is 10% of a bigger number. What stays fixed is the ratio: each value is 1.1 times the one before.

The product rule from lesson 1.4.1 says why: $b^{x+1} = b \cdot b^x$ . One more step in $x$ is one more factor of $b$ .

Rule: Properties of exponential functions

\begin{gathered} b^0 = 1 \qquad b^x > 0 \\ b^{x+1} = b \cdot b^x \end{gathered}

Every exponential passes through $(0, 1)$ and is always positive. Each step of 1 in $x$ multiplies the value by $b$ .

100 \cdot 1.1^x

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Growth by a constant percentage

If a quantity grows by a fraction $g$ at every step, each step multiplies it by $1 + g$ , as in lesson 0.3.1. After $x$ steps it has been multiplied by $(1 + g)^x$ .

Rule: Percentage growth

P(x) = P_0\,(1 + g)^x

$P_0$ is the starting amount, $g$ is the growth per step as a decimal (5% is 0.05), and $x$ is the number of steps. $P$ has the units of $P_0$ . A negative $g$ gives decay.

Try it Start at 100 and grow by 5% a step. After 10 steps the value is about 162.89, not 150. Adding 5 each time would give 150. The extra comes from growth on top of earlier growth.

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A fall works the same way. Losing 20% per step keeps 80%, so the factor is $1 - 0.2 = 0.8$ .

Check yourself

What does

P(x) = 50 \cdot 0.8^x

describe?

Growth of 80% per stepShrinking by 20% per stepShrinking by 80% per step

The shape of $b^x$ : growth and decay

Three exponentials and the flat line $1^x$ . Every curve passes through $(0, 1)$ and through $(1, b)$ . The decay curve $0.5^x$ is the mirror image of $2^x$ in the $y$ -axis.

A base above 1 rises from left to right, and a larger base rises faster. A base between 0 and 1 falls: that is decay.

No curve touches the $x$ -axis. Going left, $2^x$ halves at every step: $2^{-1} = 0.5$ , then $2^{-2} = 0.25$ , closer and closer to 0 but never 0. So the range of $b^x$ is all positive numbers (lesson 1.1.1). The $x$ -axis is a horizontal asymptote: a line the graph gets ever closer to without reaching.

Decay is growth run backwards

Since $0.5 = 2^{-1}$ , the power-of-a-power rule gives $0.5^x = 2^{-x}$ . Replacing $x$ with $-x$ reflects a graph in the $y$ -axis (lesson 1.1.2). So decay is growth read from right to left.

Rule: Negative exponents give decay

b^{-x} = \left(\frac{1}{b}\right)^x

A negative exponent on a base above 1 is the same as a positive exponent on a base below 1. So $e^{-x}$ , which you will meet often, is a decay curve.

2^x

import numpy as np import matplotlib.pyplot as plt print(" x 2^x 0.5^x") for x in range(-2, 3): print(f"{x:2} {2**x:5} {0.5**x:5}") x = np.linspace(-3, 3, 200) plt.plot(x, 2**x, label="2^x") plt.plot(x, 0.5**x, label="0.5^x") plt.axhline(0, color="gray", linewidth=0.8) plt.legend() plt.show()

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Where $e$ comes from

Suppose something grows by 100% in one step: it doubles. Now split that growth into $n$ smaller pieces of $\tfrac{1}{n}$ each, applied one after another. The total growth factor is

\left(1 + \frac{1}{n}\right)^n

With two pieces, each adds 50%: $1.5 \times 1.5 = 2.25$ , more than 2. The second piece grows the growth the first piece added.

Try it With more pieces the factor rises further, but by less each time. Going from 1 piece to 2 adds 0.25; going from 365 pieces to a million adds about 0.0037. It does not rise forever. It settles near 2.71828.

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Splitting 100% growth into $n$ pieces. The factor climbs quickly at first, then flattens just under the dashed line at $e$ , which it never passes.

The number it settles on is called $e$ , or Euler's number.

Rule: The number e

\begin{gathered} e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \\ e \approx 2.718281828 \end{gathered}

Read $\lim_{n \to \infty}$ as "the value this gets closer and closer to as $n$ grows without end". Lesson 2.1.3 makes that precise; for now the table and the figure are the evidence.

The natural exponential $e^x$

The same splitting works for any growth $x$ , not just 100%. Split it into $n$ pieces of $x/n$ and the factor $(1 + x/n)^n$ settles on $e^x$ .

It works for a negative $x$ too. With $x = -0.5$ each piece shrinks the value a little, and the factor settles on $e^{-0.5} \approx 0.606531$ : decay.

e^x

import math for x in [0.5, -0.5]: for n in [1, 10, 1000, 10**6]: print(f"x = {x}, n = {n:>7}: {(1 + x/n)**n:.6f}") print("math.exp(x) =", round(math.exp(x), 6))

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$e^x$ is an ordinary exponential with base $e$ , so every rule from lesson 1.4.1 applies.

Rule: Rules for e

\begin{gathered} e^a e^b = e^{a+b} \qquad \frac{e^a}{e^b} = e^{a-b} \\ (e^a)^b = e^{ab} \qquad e^{-x} = \frac{1}{e^x} \end{gathered}

Same rules, same reasons. $e^x$ grows, and $e^{-x}$ decays.

Common mistakes

Mixing up $x^2$ and $2^x$ . In $2^x$ the variable is the exponent. At $x = 10$ they are 100 and 1024.
Writing the percentage as a whole number. Growth of 5% multiplies by 1.05, not by 5.
Using the loss as the factor. A fall of 20% multiplies by 0.8, not by 0.2.
Adding percentages across steps. Ten steps of 5% multiply by $1.05^{10} \approx 1.6289$ , a rise of 62.89%, not 50%.
Expecting decay to reach 0. $0.5^x$ gets as small as you like but stays positive.
Reading $e^{-x}$ as a negative number. $e^{-2} \approx 0.1353$ : a negative exponent means one over.

Exponential functions and e

0 of 12 correct

1.
Simplify $e^2 \cdot e^3$
2.
Simplify $(e^x)^3$
3.
Simplify $\dfrac{e^{2x}}{e^x}$
4.
Write $\dfrac{1}{e^4}$ as a single power of $e$
5.
Evaluate $e^0$
6.
A price grows 3% per step, so its value is $200 \cdot b^x$ . What is $b$ ?
7.
A value falls 20% per step, so it is $50 \cdot b^x$ . What is $b$ ?
8.
A quantity is multiplied by 1.1 at each of 3 steps. What is the total growth factor?
9.
A value of 80 shrinks by 25% per step. What is it after 2 steps?
10.
Write $0.5^x$ as a power of 2
11.
Evaluate $\left(1 + \frac{1}{n}\right)^n$ for $n = 3$ (a fraction or a decimal)
12.
As $n$ grows, $\left(1 - \frac{1}{n}\right)^n$ settles on a power of $e$ . Which one?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

b^x

Exponential functions and the number e

From powers to functions

Adding against multiplying

Rule: Properties of exponential functions

Growth by a constant percentage

Rule: Percentage growth

The shape of $b^x$ : growth and decay

Decay is growth run backwards

Rule: Negative exponents give decay

Where $e$ comes from

Rule: The number e

The natural exponential $e^x$

Rule: Rules for e

Common mistakes

Exponential functions and e

Summary card

Free further reading

From powers to functions

Adding against multiplying

Rule: Properties of exponential functions

Growth by a constant percentage

Rule: Percentage growth

The shape of bxb^xbx: growth and decay

Decay is growth run backwards

Rule: Negative exponents give decay

Where eee comes from

Rule: The number e

The natural exponential exe^xex

Rule: Rules for e

Common mistakes

Summary card

Free further reading

The shape of $b^x$ : growth and decay

Where $e$ comes from

The natural exponential $e^x$