![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxiUY544cCu6qYy2bXhY0PZL_5y4B9kz-7KAZTQSmN-LtRHJaSShmPGO-RScfFYFa5-7Y4KP70GglCMURnYVKTtynFFxo74E6Bgof9B2Xl9bVwi2sNNXb_aCCQRMhTMefkB_qjX9xxDv-1/s320/NL.jpg)
Graph of a Natural Logarithms
Definition of Natural Logarithm
When
e y = x
Then base e logarithm of x is
ln(x) = loge(x) = y
The e constant or Euler's number is:
e ≈ 2.71828183
Common Logarithms use the Base 10Natural Logarithms use the Base e
Note:
All Natural Logarithms use the same laws and patterns as Common Logarithms
Division - Subtraction
Multiplication - Addition
Exponents - Move to the front
Same Base - Cancel
Change of Base - use e instead of 10
Natural logarithm rules
Product:
The rule name is product. The rule is,
logx (a.b) = logx a + logx b.
The example is,
logx (9.8) = logx 9 + logx 8.
Quotient:
The rule name is quotient. The rule is,
Log (a/b) = loga – logb.
The example is,
Log (10/8) = log10 – log8.
Change of base formula:
The rule name is change of base formula. The rule is
Log ab = logb /log a.
The example is,
Log 8.2 = log8 /log 2.
Power
The rule name is power. The rule is,
ln(a b) = b ∙ ln(a)
The example is,
ln(85) = 5 ∙ ln(8)
Examples
Solve for x
![]() | Divide both sides by 7 |
![]() | Use Property of Logarithms, Part 2, to take the log of both sides |
![]() | Property of Logarithms: ![]() |
![]() | ln e = 1 |
![]() | Divide both sides by 3 |
x » 1.266 |
![](http://www.occc.edu/maustin/exp_log_equations/Image859.gif)
![]() | |
![]() | Use Property of Logarithms, Part 2, to take the log of both sides |
(x + 2) ln 2 = (2x + 1) ln 3 | Property of Logarithms: ![]() |
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3 | Distributive Property |
x ln 2 - 2x ln 3 = ln 3 – 2 ln 2 | Isolate terms with the variable on one side of the equation |
x(ln 2 – 2 ln 3) = ln 3 – 2 ln 2 | Factor out the common factor, x |
x » 0.191 |