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Logarithm Concepts -- Introduction

Module by: Kenny M. Felder. E-mail the author

Summary: This module introduces the concept of logarithms.

Suppose you are a biologist investigating a population that doubles every year. So if you start with 1 specimen, the population can be expressed as an exponential function: p(t)=2tp(t)=2t size 12{p \( t \) =2 rSup { size 8{t} } } {} where tt size 12{t} {} is the number of years you have been watching, and pp size 12{p} {} is the population.

Question: How long will it take for the population to exceed 1,000 specimens?

We can rephrase this question as: “2 to what power is 1,000?” This kind of question, where you know the base and are looking for the exponent, is called a logarithm.

log21000log21000 size 12{"log" rSub { size 8{2} } "1000"} {} (read, “the logarithm, base two, of a thousand”) means “2, raised to what power, is 1000?”

In other words, the logarithm always asks “What exponent should we use?” This unit will be an exploration of logarithms.

A few quick examples to start things off

Table 1
Problem Means The answer is because
log 2 8 log 2 8 2 to what power is 8? 3 2 3 2 3 is 8
log 2 16 log 2 16 2 to what power is 16? 4 2 4 2 4 is 16
log 2 10 log 2 10 2 to what power is 10? somewhere between 3 and 4 2 3 = 8 2 3 =8 and 2 4 = 16 2 4 =16
log 8 2 log 8 2 8 to what power is 2? 1 3 1 3 8 1 3 = 8 3 = 2 8 1 3 = 8 3 =2
log 10 10,000 log 10 10,000 10 to what power is 10,000? 4 10 4 = 10,000 10 4 =10,000
log 10 ( 1 100 ) log 10 ( 1 100 ) 10 to what power is 1 100 1 100 ? –2 10 –2 = 1 10 2 = 1 100 10 –2 = 1 10 2 = 1 100
log 5 0 log 5 0 5 to what power is 0? There is no answer 5 something 5 something will never be 0

As you can see, one of the most important parts of finding logarithms is being very familiar with how exponents work!

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