1.4.3 Logarithms and their rules

A logarithm is an exponent

Two to what power gives 8? The answer is 3, because $2^3 = 8$ . That answer has a name: the logarithm of 8 to base 2, written $\log_2 8 = 3$ .

Rule: Definition of a logarithm

\begin{gathered} \log_b y = x \quad \text{means} \quad b^x = y \\ (b > 0,\ b \neq 1,\ y > 0) \end{gathered}

A logarithm answers one question: "what exponent?" Every log statement is an exponent statement written the other way round.

A log counts factors. $\log_2 8 = 3$ because 8 is three factors of 2: three doublings, starting from 1. $\log_{10}$ counts factors of 10, so $\log_{10} 1000 = 3$ . A number between 1000 and 10,000 has a log between 3 and 4: $\log_{10} 5000 \approx 3.699$ .

A log is a count, so it has no units.

Exponential form	Log form	In words
$2^3 = 8$	$\log_2 8 = 3$	2 to the power 3 is 8
$10^3 = 1000$	$\log_{10} 1000 = 3$	10 to the power 3 is 1000
$5^0 = 1$	$\log_5 1 = 0$	any base to the power 0 is 1
$2^{-2} = \tfrac{1}{4}$	$\log_2 \tfrac{1}{4} = -2$	a negative log means a number below 1

Try it Python has math.log2 and math.log10 for the two common bases. For any other base, give it as a second input: math.log(y, b) . Check each row of the table.

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Logs of numbers below 1

A log can be negative; its input cannot. Divide by 10 and the base-10 log drops by 1:

$y$	1000	100	10	1	0.1	0.01	0.001
$\log_{10} y$	3	2	1	0	$-1$	$-2$	$-3$

The algebra agrees: $0.01 = 10^{-2}$ (lesson 1.4.1), so $\log_{10} 0.01 = -2$ . So with a base above 1, a number above 1 has a positive log and 1 has log 0. A number between 0 and 1 has a negative log.

Keep dividing and the log keeps falling, without end. But no power of 10 is ever 0 or negative, because $10^x$ is always positive (lesson 1.4.2). So $\log_b y$ exists only for $y > 0$ .

-5

import math for y in [1000, 100, 10, 1, 0.1, 0.01, 0.001]: print(y, math.log10(y)) for y in [0, -5]: try: print(math.log10(y)) except ValueError as err: print(y, "has no log:", err)

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The natural logarithm

The logarithm with base $e$ has its own name and symbol: the natural log, $\ln y = \log_e y$ . It asks " $e$ to what power gives $y$ ?"

Rule: Logs and exponentials undo each other

\begin{gathered} \ln\left(e^{x}\right) = x \qquad e^{\ln y} = y \\ \ln e = 1 \qquad \ln 1 = 0 \end{gathered}

Apply one and then the other and you are back where you started. The same holds for any base: $\log_b(b^x) = x$ and $b^{\log_b y} = y$ .

$e^x$ always rises, so no two inputs share an output: it passes the horizontal line test of lesson 1.1.3. So it has an inverse function, and that inverse is $\ln$ . Its graph is the graph of $e^x$ mirrored in the line $y = x$ .

$e^x$ and $\ln x$ are mirror images in $y = x$ : each point $(a, b)$ on one has the partner $(b, a)$ on the other. The pair $(-1, \frac{1}{e})$ and $(\frac{1}{e}, -1)$ shows a number below 1 with a negative log.

Mirroring swaps the domain and the range. $e^x$ takes any input and gives a positive output, so $\ln x$ takes a positive input and gives any output. It also rises slowly: $\ln 1000 \approx 6.9078$ .

\ln y

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The three log rules

Logs turn multiplication into addition. Here are the first powers of 2 with their logs:

$y$	1	2	4	8	16	32
$\log_2 y$	0	1	2	3	4	5

Multiplying in the top row is adding in the bottom row: $4 \times 8 = 32$ , and $2 + 3 = 5$ . The product rule of exponents (lesson 1.4.1) is the reason:

\begin{aligned} 8 \times 4 &= 2^3 \times 2^2 = 2^{3+2} = 32 \\ \text{so}\quad \log_2 (8 \times 4) &= 3 + 2 \\ &= \log_2 8 + \log_2 4 \end{aligned}

Each exponent rule from lesson 1.4.1 gives one log rule in the same way.

Rule: Product rule

\log_b(xy) = \log_b x + \log_b y

The log of a product is the sum of the logs.

Rule: Quotient rule

\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y

The log of a quotient is the difference of the logs.

Rule: Power rule

\log_b\left(x^k\right) = k \log_b x

An exponent inside a log comes down in front as a multiplier. This is the rule you will use most: it brings an unknown exponent down where you can solve for it (lesson 1.4.4).

Two consequences are worth naming. The quotient rule with $\log_b 1 = 0$ gives $\log_b \frac{1}{x} = -\log_b x$ . And the product rule works for many factors at once, turning a product $\Pi$ into a sum $\Sigma$ (lesson 0.2.2):

\begin{gathered} \ln(x_1 x_2 \cdots x_n) \\ = \ln x_1 + \ln x_2 + \cdots + \ln x_n \end{gathered}

Case study 1.4.7 uses this to add up returns over time. All the rules need $x > 0$ and $y > 0$ . If you want to see why they must hold in general, Proof Gym stage PG2 proves them.

Try it Each line prints both sides of one rule, using natural logs. The last line is the product rule with three factors. The pairs agree, apart from rounding in the last digits.

import math print(math.log(6), math.log(2) + math.log(3)) print(math.log(100) - math.log(25), math.log(4)) print(math.log(2**10), 10 * math.log(2)) print(math.log(1.02 * 0.97 * 1.05), math.log(1.02) + math.log(0.97) + math.log(1.05))

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Two worked examples

To expand $\ln\dfrac{x^3}{\sqrt{y}}$ , split it with the quotient rule. Then write $\sqrt{y} = y^{1/2}$ (lesson 1.4.1) so the power rule can bring both exponents down:

\begin{aligned} \ln\frac{x^3}{\sqrt{y}} &= \ln x^3 - \ln y^{1/2} \\[4pt] &= 3\ln x - \tfrac{1}{2}\ln y \end{aligned}

To combine $2\ln 6 - \ln 9$ into one log, run the rules the other way. The power rule turns $2\ln 6$ into $\ln 36$ , then the quotient rule joins the two logs:

\begin{aligned} 2\ln 6 - \ln 9 &= \ln 36 - \ln 9 \\[4pt] &= \ln\frac{36}{9} = \ln 4 \end{aligned}

x = 2

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Change of base

Python and calculators give $\ln$ directly. For any other base, divide two natural logs.

Rule: Change of base

\log_b y = \frac{\ln y}{\ln b}

So $\log_2 10 = \dfrac{\ln 10}{\ln 2} \approx 3.3219$ , which means $2^{3.3219} \approx 10$ .

It works for a base below 1 too: $\log_{0.5} 8 = \frac{\ln 8}{\ln 0.5} = -3$ , because $0.5^{-3} = 8$ . A base of 1 is impossible: $\ln 1 = 0$ , and the formula would divide by zero.

Any power as a power of $e$

Since $b = e^{\ln b}$ , the power-of-a-power rule gives a second form of change of base.

Rule: Powers through e

b^x = e^{x \ln b}

Every exponential is $e^x$ with $x$ stretched by the factor $\ln b$ . For example $2^x = e^{0.693147x}$ , since $\ln 2 \approx 0.693147$ . Lesson 2.3.4 uses this to differentiate $b^x$ .

\log_2 10

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Mistakes to avoid

\ln(2 + 3) = \ln 5 \approx 1.609

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Logarithms

0 of 11 correct

1.
Evaluate $\log_2 32$
2.
Evaluate $\log_{10} 0.01$
3.
Evaluate $\log_7 1$
4.
Evaluate $\ln\left(e^{4}\right)$
5.
Write $\ln 5 + \ln 3$ as a single logarithm
6.
If $\ln x = a$ and $\ln y = b$ , write $\ln\left(x^3 y\right)$ in terms of $a$ and $b$
7.
If $\ln x = a$ and $\ln y = b$ , write $\ln\left(\dfrac{x}{y^2}\right)$ in terms of $a$ and $b$
8.
If $\ln x = a$ , write $\ln\left(\dfrac{1}{x}\right)$ in terms of $a$
9.
Use change of base to find $\log_2 10$ (exactly, or to 4 significant figures)
10.
Evaluate $\log_{1/2} 8$
11.
Write $3^x$ as a power of $e$

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.

\log_b y = x

Logarithms and their rules

A logarithm is an exponent

Rule: Definition of a logarithm

Logs of numbers below 1

The natural logarithm

Rule: Logs and exponentials undo each other

The three log rules

Rule: Product rule

Rule: Quotient rule

Rule: Power rule

Two worked examples

Change of base

Rule: Change of base

Any power as a power of $e$

Rule: Powers through e

Mistakes to avoid

Logarithms

Summary card

Free further reading

A logarithm is an exponent

Rule: Definition of a logarithm

Logs of numbers below 1

The natural logarithm

Rule: Logs and exponentials undo each other

The three log rules

Rule: Product rule

Rule: Quotient rule

Rule: Power rule

Two worked examples

Change of base

Rule: Change of base

Any power as a power of eee

Rule: Powers through e

Mistakes to avoid

Summary card

Free further reading

Any power as a power of $e$