Basic Logarithms A way to Undo exponents
Many things we do in mathematics involve undoing an operation.
Subtraction is the inverse of addition
When you were in grade school, you probably learned about subtraction this way. 2 + = = 10
Then one day your teacher introduced you to a new symbol ─ to undo addition
3 + = 10 Could be written 10 ─ 3 =
8 – 2 =
2 + ? = 8
8 – 2 = = 8
8 – 2 = = 8
The same could be said about division ÷
40 ÷ 5 =
5 x ? = 40
40 ÷ 5 = 5 x 8 = 40
40 ÷ 5 = 8 5 x 8 = 40
Consider √49
= ?
? 2 = 49
= ? 7 2 = 49
= = 49
Exponential Equations: 5 ? = 25
Exponential Equations: 5 2 = 25
Logarithmic Form of 5 2 = 25 is log 5 25 = 2
log 5 25 = ?
5 ? = 25
log 5 25 = ? 5 2 = 25
log 5 25 = = 25
Try this one…
log 7 49 = ?
7 ? = 49
log 7 49 = ? 7 2 = 49
log 7 49 = = 49
and this one…
log 3 27 = ?
3 ? = 27
log 3 27 = ? 3 3 = 27
log 3 27 = = 27
Remember your exponent rules? 7 0 = ? 5 0 = ?
Remember your exponent rules? 7 0 = = 1
log 7 1 = ?
7 ? = 1
log 7 1 = ? 7 0 = 1
log 7 1 = = 1
Keep going…
log 3 1 = ?
3 ? = 1
log 3 1 = ? 3 0 = 1
log 3 1 = = 1
Remember this? 1/25 = 1/ 5 2 = 5 -2
log 5 ( )= ?
5 ? = 1/25 log 5 ( )= ?
5 -2 = 1/25 log 5 ( )= ?
5 -2 = 1/25 log 5 ( )= -2
Try this one…
log 3 ( )= ?
3 ? = log 3 ( )= ?
3 -4 = log 3 ( )= ?
3 -4 = log 3 ( )= -4
Let’s learn some new words. When we write log is called the base 125 is called the argument
When we write log 2 8 The base is ___ The argument is ___
When we write log 2 8 The base is 2 The argument is 8
Back to practice…
log =?
10 ? =1000
log =? 10 3 =1000
log = =1000
And another one
log 10 ( )=?
10 ? =
log 10 ( )=? =
log 10 ( )= =
log 10 is used so much that we leave off the subscript (aka base)
log can be written log 100
log =?
10 ? =10000
log =? 10 4 =10000
log = =10000
And again
log 10 = ?
10 ? =10
log 10 = ? 10 1 =10
log 10 = =10
What about log 33?
What about log 33? We know 10 1 = 10 and 10 2 = 100 since 10 < 33 < 100 we know log 10 < log 33 < log 100
Add to log 10 < log 33 < log 100 the fact that log 10 = 1 and log 100 = 2 to get 1 < log 33 < 2
A calculator can give you an approximation of log 33. Look for the log key to find out… (okay, get it out and try)
log 33 is approximately
Guess what log 530 is close to.
100 < 530 < 1000 so log 100 < log 530 < log 1000 and thus 2 < log 530 < 3
Your calculator will tell you that log 530 ≈ ….
Now for some practice with variables. We’ll be solving for x.
log 4 16 = x
log 4 16 = x 4 ? = 16
log 4 16 = x 4 2 = 16
log 4 16 = x x=2 4 2 = 16
Find x in this example.
log 8 x = 2
log 8 x = = ?
log 8 x = = 64
log 8 x = 2 x= = 64
Find x in this example.
log x 36 = 2
log x 36 = 2 x 2 = ?
log x 36 = 2 x 2 = 36
log x 36 = 2 x= 6 x 2 = 36
We need some rules since we want to stay in real number world. Consider log base (argument) = number The base must be > 0 The base cannot be 1 The argument must be > 0
Why can’t the base be 1? 1 4 = =2 That would mean log 1 1=4 Log 1 1=10 That would be ambiguous, so we just don’t let it happen.
Why must the argument be > 0? 5 2 =25 and 25 is positive 5 0 =1 and 1 is positive 5 -2 = 1/25 and that’s positive too Since 5 to any power gives us a positive result, the argument has to be a positive number.