Logarithm

Bases and Exponents

A power function has the form

y = ax

a is known as the base, while x is called the power or exponent, and the power function means that a is multiplied by itself x times:

	y(x)	=	ax
		=	a·a·····a
			x times	.

Example:

	y(3)	=	23
		=	2·2·2
		=	8	.

We often say that a is raised to the power x.

§1.2 Combinations of Bases and Powers

Some simple combinations of bases and powers illustrate the utility of power functions.

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§1.2.1 Powers of Products

If the base of a power function is a product of two numbers, (a·b)^x, multiplying out and rearranging the terms tells us that the result can be expressed as the the product of two powers:

(a·b)x = ax·bx

Explanation:

	(a·b)x	=	(a·b·a·b·····a·b)
			x times
			(a·a·····a)	·	(b·b·····b)
			x times		x times
		=	ax·bx			.

Example:

	(2·3)2	=	22·32
	62	=	4·9
	36	=	36	.

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§1.2.2 Multiplication of Power Functions

When power functions with the same base are multiplied together, a^w·a^x, that base is multiplied by itself w times, and then again by x times, which is equivalent to adding their exponents:

aw·ax = aw+x

Explanation:

	aw·ax	=	(a·a·····a)	·	(a·a·····a)
			w times		x times
		=	(a·a·a·a·····a·a)
			w + x times
		=	aw+x			.

Example:

	24·23	=	(2·2·2·2)·(2·2·2)
		=	27	.

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§1.2.3 Division of Power Functions

When power functions with the same base are dividing each other, a^w/a^x, that base is multiplied by itself w times, and then divided by the base multiplied by itself x times, which is equivalent to subtracting their exponents:

aw/ax = aw–x

Explanation:

	aw / ax	=	(a·a·····a)	/	(a·a·····a)
			w times		x times
		=	(a·a·a·a·····a·a)
			w – x times
		=	aw–x			.

Example:

	24 / 23	=	(2·2·2·2) / (2·2·2)
		=	21	.

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§1.2.4 Powers of Powers

If a power function is itself a base of another power, (a^w)^x, the base a is multiplied by itself w times, and the result is then multiplied together x times, which is equivalent to multiplying their exponents:

(aw)x = aw·x

Explanation:

	(aw)x	=	(a·a·····a)	·	(a·a·····a)	·····	(a·a·····a)
			w times		w times		w times
			x times
		=	(a·a·a·a·····a·a)
			w · x times
		=	aw·x					.

Example:

	(24)3	=	(2·2·2·2)·(2·2·2·2)·(2·2·2·2)
		=	27	.

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§1.3 Particular Exponent Values

The previous examples have implied that exponents were positive integers. But all types of exponents are possible.

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§1.3.1 Special Exponents of 0 and 1

One special case is the exponent of 1, which means that the base a is multiplied by itself just one time:

a1 = a

Another special case is the exponent of 0, which means that there are no multiplications of the base at all:

a0 = 1

Another way to see that a⁰ is equal to 1 is to divide any power function by itself:

	ax/ax	=	ax–x
	1	=	a0	.

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§1.3.2 Negative Exponents

We see from above that decreasing the exponent is equivalent to dividing by the base. Eventually we must reach a number less than one, so it shouldn't be a surprise to understand a negative exponent as being 1 divided by the base raised to the positive exponent:

a–x = 1/ax

Explanation:

	a–1	=	a0–1
		=	a0/a1	Why?
		=	1/a	Why?
	a–x	=	(a–1)x	Why?
		=	(1/a)x
		=	1x/ax	Why?.
		=	1/ax	Why?.

Example:

	2–3	=	1/(2·2·2)
		=	1/8	.

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§1.3.3 Rational Exponents

Suppose an exponent in a power function y = a^x is a non-integer rational value, e.g. a fraction such 2/3.
Let's first consider a rational exponent of the form 1/n, y = a^1/n; since multiplying it together n times must result in just a, this must be the n^th root of a:

a1/n =

Explanation:

	(a1/n)n	=	an/n	Why?
		=	a1
		=	a	Why?.

Example:

	(41/2)2	=	41
	(±2)2	=	4	.

Now consider a more general rational exponent y = a^m/n:

am/n = ()m

So, any base a raised to a rational exponent can be expressed as an n^th root of a raised to an integer power.

Example:

	163/4	=	(161/4)3
		=	(±2)3 or (±2 i)3
		=	±8 or ±8 i	.

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§1.3.4 Real Exponents

If the exponent in a power function y = a^x is an integer, y can be calculated in a straightforward fashion:

	y(2)	=	42
		=	4·4
		=	16	;
	y(3)	=	43
		=	4·4·4
		=	64	.

Intermediate values of exponents that are rational can be calculated as described above:

	y(2.5)	=	42.5
		=	42+1/2
		=	42 · 41/2	Why?
		=	16 · ±2	Why?
		=	±32	.

In general, though, any value of a and x are possible, and a scientific calculator may be required to easily evaluate them:

	y(2.56789)	=	4.12.56789
		=	37.459...

As you might expect, this value lies between the previous two (positive) values, 32 and 64.
Note that, while roots can be negative or complex, a scientific calculator will only return a real and preferably positive root (though that's generally what you want).
Scientific calculators are able to calculate these functions by approximating them with a sum of rational values.

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§1.4 Particular Base Values

Some values of the base of a power function have particular use and importance.

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§1.4.1 Special Base 1

The base 1 is special because it is unaffected by any exponent, since multiplying by 1 never changes a value:

1x = 1

Explanation:

	1x	=	1·1·····1
			x times
		=	1	.

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§1.4.2 Base 10 and Scientific Notation

Because our number system is based on the number 10, it is often convenient to express very large and very small numbers as a value w between 1 and 10 (the prefactor) multiplied by a power of 10:

y = w · 10x

Examples:

	37.46	=	3.746 · 101
	1249	=	1.249 · 103
	521,700	=	5.217 · 105
	0.02836	=	2.836 · 10–2	.

This format allows us to first consider the "order of magnitude", i.e. the power of 10, and then the specific value in that range of numbers, i.e. the prefactor.
Scientific calculators have a special key to enter the base-10 exponent after the prefactor, often labeled EXP or EE (for "Enter Exponent").
When displaying such numbers, calculators and computer programming languages will often drop the "· 10", writing them in a format such as "5.217⁰⁵" or " 5.217e05".

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§1.4.3 The Base e and the Exponential Function

One particular base is of singular importance in math and science: the transcendental number

e = 2.718281...

The base e shows up naturally in calculus because the power of this base is equal to its own derivative:

	y	=	ex
	dy/dx	=	ex	.

The base-e power function, e^x, is called the exponential function, and it is sometimes abbreviated exp:

ex

=

exp(x)

.

(This abbreviation should not be confused with the calculator key EXP, if that's used by your calculator to enter exponents.)
For reasons we'll see later, any power function can be expressed as a power of the exponential function:

ax

=

ex/l

,

where

l

=

1 / ln a

.

Therefore power functions in general are sometimes also referred to as exponential functions.
The scale l describes the amount of change in x required for a change in the function by a factor of e, which is a common standard of comparison.

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§1.4.4 Negative Bases

Negative bases can always be separated into factors of -1 and a positive base:

(–a)x = (–1)x·ax

Unless the exponent x is an integer, the factor (–1)^x will likely be a complex number, e.g.

(–1)1/2

=

i

.

Full discussion of such bases is beyond the scope of this tutorial.

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§1.5 Graphs of Power Functions

Since we have defined power functions for all real exponents, we can draw graphs of them to see how they behave in general:

Note the special base a = 1, as well as the function values at the special exponents x = 0 and 1.
Note also the left-right symmetry of 2^x and 2^–x.
This graph should help you understand why the term "exponential" is so commonly used to describe a rapid change, either an increase or a decrease.

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§2. Logarithms

Many natural phenomena vary as a power function a^x, so that they display exponential change over widely measurable ranges of the exponent x. Scientists are therefore often interested in the value of this exponent, which is called a logarithm.

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§2.1 Logarithms and Powers

Quite often when analyzing a power function y = a^x , it is y that is the known quantity, and the exponent x is the value that one wishes to find.
One then needs a function x = f(y) that is the inverse of the power function, which is called a logarithm:

x = log a y

y is called the argument of the logarithm.
Another way of saying this is that x is the exponent to which a must be raised to equal y:

x

=

log a y

if and only if

ax

=

y

.

When going from a logarithmic relation to a power relation, it helps to visually imagine the "log" disappearing as you bring the base a to the other side of the equality and raise it to the power x:

Note that we can always very generally write:

y

=

aloga y

,

which emphasizes that a logarithm is just an exponent of a power function.

Examples:

	x	=	log 2 8
	2x	=	8 = 23
	x	=	3	.

	4y	=	17
	y	=	log 4 17	.

Exercises:

Type 1: Find an expression for y given that x = log _a y [substitute various symbols for {a, x, y}, as well as various symbols and simple numerical values for {a, x} — no ones or zeros].
Type 2: Calculate x = log _a y for {a, y} [substitute simple numerical values such as {2, 16}, {3, 27}, {1/9, 1/81}, etc. — no ones or zeros]

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§2.2 Combinations of Logarithms

Some simple combinations of logarithms illustrate their utility.

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§2.2.1 Logarithms of Products

When the argument of a logarithm is a product of two numbers, log _a (y · z), it can be split apart into the sum of two logarithms:

log a (y · z) = log a y + log a z

Explanation:

	x	=	log a (y · z)
	ax	=	y · z	Why?
		=	aloga y · aloga z	Why?
		=	aloga y + loga z	Why?
	x	=	log a y + log a z	Why? .

Examples:

	x	=	log 5 (2·3)
		=	log 5 2 + log 5 3	.

	x	=	log a (y · z)
	ax	=	y · z	Why?
	y	=	ax/z	.

	x	=	aloga y + loga z
		=	aloga y· aloga z	Why?
		=	y · z	Why? .

Exercises:

Type 1: Find an expression for x = log _a (y · z) that involves logarithms of single arguments [substitute various symbols for {a, y, z}, as well as various simple numerical values for {a, y, z} — no ones or zeros].
Type 2: Find an expression for y given that x = log _a (y · z) [substitute various symbols for {a, y, z}].
Type 3: Find a more compact expression for x = log _a y + log _a z [substitute various symbols for {a, y, z}, as well as various simple numerical values for {a, y, z} — no ones or zeros].
Type 4: Find a more compact expression for x = a^{log_a y + log_a z} that doesn't use any logarithms [substitute various symbols for {a, y, z}, as well as various simple numerical values for {a, y, z} — no ones or zeros].

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§2.2.2 Logarithms of Quotients

When the argument of a logarithm is a quotient of two numbers, log _a (y / z), it can also be split apart, into the difference of two logarithms:

log a (y / z) = log a y – log a z

Explanation:

	x	=	log a (y / z)
	ax	=	y / z	Why?
		=	aloga y / aloga z	Why?
		=	aloga y – loga z	Why?
	x	=	log a y – log a z	Why? .

Examples:

	x	=	log 5 (2/3)
		=	log 5 2 – log 5 3	.

	x	=	log a (y/z)
	ax	=	y/z	Why?
	y	=	z · ax	.

	x	=	aloga y – loga z
		=	aloga y/ aloga z	Why?
		=	y / z	Why? .

Exercises:

Type 1: Find an expression for x = log _a (y / z) that involves logarithms of single arguments [substitute various symbols for {a, y, z}, as well as various simple numerical values for {a, y, z} — no ones or zeros].
Type 2: Find an expression for x = log _a (w · y / z) that involves logarithms of single arguments [substitute various symbols and simple numerical values for {a, w, y, z} — no ones or zeros].
Type 3: Find an expression for z given that x = log _a (y / z) [substitute various symbols and simple numerical values for {a, y} — no ones or zeros].
Type 4: Find a more compact expression for x = a^{log_a y – log_a z} that doesn't use any logarithms [substitute various symbols and simple numerical values for {a, y, z} — no ones or zeros].

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§2.2.3 Logarithms of Power Functions

When the argument of a logarithm is a power of two numbers, log _a b^y, it can also be split apart, into a multiple of a logarithm:

log a by = y log a b

Explanation:

	x	=	log a by
		=	log a (b·b·····b)		Why?
			y times
		=	log a b + log a b+ ... + log a b		Why?
			y times
		=	y log a b		.

Example:

	x	=	log 5 8
		=	log 5 23
		=	3 log 5 2	.

	x	=	log a (z · by)
		=	log a z + log a by	Why?
x – log a	z	=	y log a b
	y	=	(x – log a z)/log a b	.

	x	=	w log a y + log a z
		=	log a yw + log a z
		=	log a (yw · z)	Why? .

	x	=	log a (by / bz)
		=	log a by–z	Why?
		=	(y – z) log a b	.

Exercises:

Type 1: Find an expression for x = log _a b^y that doesn't use any exponents [substitute various symbols for {a, b, y}, as well as various simple numerical values for {a, b, y} — no ones or zeros].
Type 2: Find an expression for y given that x = log _a (b^y · c^z) [substitute various symbols for {a, b, c, y, z}, as well as various simple numerical values for {a, b, c, y, z} — no ones or zeros].
Type 3: Find an expression for x = v log _a w + y log _a z that involves only a single logarithm function [substitute various symbols for {a, v, w, y, z}, as well as various simple numerical values for {a, v, w, y, z} — no ones or zeros].

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§2.3 Particular Argument Values

Some arguments of the logarithm function have special characteristics.

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§2.3.1 Special Arguments

Because any number raised to the power of 1 is equal to itself, a¹ = a, the following identity will always be true for any argument equal to its base:

log a a = 1

Similarly, because any number raised to the power of 0 is equal to 1, a⁰ = 1, the following identity will always hold for any base a:

log a 1 = 0

Because a power function a^x only reaches a value of 0 for very large exponents x → – ∞ (a > 1) and x → + ∞ (a < 1), an argument of zero will result in:

log a 0 = ± ∞ (a ≶ 1)

Logarithms of negative arguments generally result in complex values, and their discussion is beyond the scope of this tutorial.

Examples:

	x	=	log 5 log 3 3
		=	log 5 1
		=	0	.

	x	=	log a (ay / az)
		=	log a ay – log a az	Why?
		=	y log a a – z log a a	Why?
		=	y – z	.

	x	=	log 5 log 3 1
		=	log 5 0
		=	– ∞	.

Exercises:

Type 1: Calculate x = log _b log _a a [substitute various symbols for {a, b}, as well as various simple numerical values for {a, b} — no ones or zeros].
Type 2: Calculate x = log _b log _a 1 [substitute various symbols for {a, b}, as well as various simple numerical values for {a, b} — no ones or zeros].
Type 3: Calculate x = log _a (a^y / a^z) [substitute various symbols for {a, b}, as well as various simple numerical values for {a, b} — no ones or zeros].

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§2.4 Particular Base Values

Some values of the base of a logarithm have particular use and importance.

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§2.4.1 Common Logarithms

In science, large and small numbers are often expressed in scientific notation, e.g.

	1,000,000	=	106	,
	3,512,000	=	3.512 x 106	.

For this reason powers of 10 are very important, and we commonly want to calculate the exponent on such a power, e.g.

	x	=	log 10 1,000,000
		=	6	.

and

	x	=	log 10 3,512,000
		=	6.5455545...

Again a scientific calculator may be required to evaluate such expressions (note that the log₁₀ key is usually implemented as inv 10^x, or vice-versa).
The base-10 logarithm is called the common logarithm, and the subscript is typically dropped:

x = log 10 y = log y

Exercises:

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§2.4.2 Natural Logarithms

Another base that shows up naturally in calculus is the transcendental number e = 2.718281..., because the power of this base is equal to its own derivative.
The base-e logarithm, the inverse of e^x, is called the natural logarithm, and it is typically abbreviated ln:

x = log e y = ln y

This function can also be found on a scientific calculator (and again note that the ln key is usually implemented as inv e^x, or vice-versa).
The exponential function can be used to define any other power function, because any base can be expressed as a power of e with a scaled exponent:

ax = ex · ln a

Explanation:

	a	=	eln a	Why?
	ax	=	(eln a)x
		=	ex · ln a	Why? .

Exercises:

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§2.5 Graphs of Logarithm Functions

Since we have defined logarithm functions for all real positive arguments, we can draw graphs of them to see how they behave in general:

Note the special base a = 1, as well as the function values at the special arguments x = 1 and x = a.
Note also the up-down symmetry of log ₂ x and log _0.5 x.

This graph should help you understand why logarithmic growth is commonly used to describe a slow increase.

http://webmath.amherst.edu/qcenter/logarithms/index.html

https://www.youtube.com/watch?v=eLapHtvQbFo