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.
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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