Introduction to Logarithms

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INTRODUCTION TO LOGARITHMS

Jump right to the exercises!

Before doing this exercise, you may want to review exponents and function notation:
Multi-Step Exponent Law Practice
Introduction to Function Notation

There are two basic views of logarithms that you must know.

One view is that logarithmic functions "undo" exponential functions.
This view requires a knowledge of exponential functions, one-to-one functions, and inverse functions;
these skills will be developed in the next few sections.

The other view is more basic, and is a good way to start thinking about logarithms: logarithms are exponents.
This viewpoint is introduced in this section.

Logarithms are Numbers with Ugly Names

Logarithms (commonly shortened to just logs) are just numbers.
Part of the problem is that—well—they have ugly names, like:

log₂ 8

log₃ 9

lne

log 1000

You need to learn how to read these numbers aloud,
and to understand the property that each of these numbers satisfies.

First of all, recall function notation:
if the name of a function is f , and the input to the function is x ,
then the unique corresponding output is called f(x)
(which is read as " f of x ".)

The number log₂ 8 is the function log₂ (read aloud as log base 2 ) acting on the input 8 .
The number log₃ 9 is the function log₃ (read aloud as log base 3 ) acting on the input 9 .

There is a convention in mathematics about functions that have more than one letter to their name (like log₂ or log₃ ):
if there's no confusion about the order of operations, then you usually DROP the parentheses that are usually used to "hold" the input.
That is, it is conventional to write log₂ 8 , instead of log₂(8) .

The function f only has one letter; you would never shorten f(x) to fx , because this could be confused with multiplication.
However, when the function name is familiar and has more than one letter, then there's no potential for confusion.

You read log₂ 8 aloud using normal function notation: log base 2 of 8.
(Notice the word of which denotes the function acting on the input!)

It's very important that you understand that log₂ 8 is the output from the function log₂ when the input is 8 .

Two Special Logarithms

You'll learn more about the bases of logarithms (the little number in the subscript position) later on.
For now, however, you should know that there are two important logarithms that are given special names:

COMMON LOGARITHM (log)

The function log₁₀ (log base 10)
is called the common logarithm or common log
and is abbreviated by just log .

That is, when you see the function log with NO base, the base is actually 10 .

The number log 1000 is read aloud as the common log of 1000.

NATURAL LOGARITHM (ln)

The function log_e (log base e; that's the irrational number e , which is approximately 2.72)
is called the natural logarithm or natural log
and is abbreviated by just ln .

That is, when you see the function ln , you must recognize this as the logarithm with base e .

The number lne is read aloud as the natural log of the square root of e .

The common log and the natural log are the only two logs that appear on many handheld calculators.

Logarithms are Exponents

Question: So, what exactly IS the number log₂ 8 ?
Answer: It is the POWER that 2 must be raised to, to get 8 !
It is convenient to think of making the "circle" shown below:

Since 2³ = 8 , it follows that log₂ 8 = 3 .
(Read this aloud as: log base 2 of 8 is 3 )

Thus, we have the KEY IDEA:

LOGARITHMS ARE EXPONENTS
or
LOGARITHMS ARE POWERS

EXAMPLES

What is log₃ 9 ?    It is the power that 3 must be raised to, to get 9 .
Since 3² = 9 , it follows that log₃ 9 = 2 .

What is log 1000 ?    It is the power that 10 must be raised to, to get 1000 .
Since 10³ = 1000 , it follows that log 1000 = 3 .

What is lne ?    It is the power that e must be raised to, to get e .
Since e12=e , it follows that lne=12 .

What is log₅₇ 157 ?    It is the power that 57 must be raised to, to get 157 .
Since 57^-1 = 157 , it follows that log₅₇ 157 = -1 .

What is log₂₄ 1 ?    It is the power that 24 must be raised to, to get 1 .
Since 24⁰ = 1 , it follows that log₂₄ 1 = 0 .

ESTIMATE the following logarithm:   log₃ 7
That is, find two positive integers between which the logarithm lies.
Give a reason to support your answer.
Answer: Since 3¹ = 3 and 3² = 9 , log₃ 7 lies between 1 and 2 .

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.