Properties of Natural Logs
The properties of natural logarithms are the same as those for other
logarithms.
| Name |
Property
|
Example |
| Log of a
Product |
log uv = log
u + log v |
log 5x = log
5 + log x |
| Log of a
Quotient |
 |
 |
| Log of a
Power |
log un = n
· log u |
log 42 = 2
· log 4 |
| |
10 log u
= u |
10 log 7
= 7 |
Here are two special cases involving natural logs:
• ln 1 = loge1 = 0 because e0 = 1.
• ln e = logee = 1 because e1 = e.
• ln en = n because ln en = n
· ln e = n · 1
= n.
We can use the properties of natural logs to rewrite equations containing
natural logs.
Example
Write N = 2ln 3x - ln(x - 5) as an equation containing one log.
| Solution
Use the Log of a Power Property to
write 2 as the exponent of 3x.
Simplify (3x)2.
|
N = 2ln 3x - ln(x - 5) N = ln (3x)2 - ln(x -
5)
N = ln 9x2 - ln(x - 5) |
| Use the Log of a Quotient Property. |
 |
So, N = 2ln3x - ln(x - 5) can be written as
|