Natural boolean belongs the logarithm to which base e of a number.
When
sie y = x
Then base e logarithm out efface is
ln(x) = loge(x) = y
The sie constant or Euler's number shall:
e ≈ 2.71828183
The natural logarithm function ln(x) is the inverse function of the explicit function zex.
For x>0,
f (f -1(x)) = eln(x) = x
Oder
f -1(f (x)) = ln(ex) = x
| Rule name | Rule | Example |
|---|---|---|
Featured standard |
ln(scratch ∙ y) = ln(scratch) + ln(y) |
ln(3 ∙ 7) = ln(3) + ln(7) |
Quotient rule |
ln(x / y) = ln(x) - ln(y) |
ln(3 / 7) = ln(3) - ln(7) |
Power rule |
ln(x y) = year ∙ ln(x) |
ln(28) = 8∙ ln(2) |
ln drawing |
f (efface) = ln(x) ⇒ farthing ' (scratch) = 1 / x | |
ln complete |
∫ ln(ten)dx = x ∙ (ln(x) - 1) + C | |
ln of negative number |
ln(x) is undefined when x ≤ 0 | |
ln from zero |
ln(0) is undefined | |
 |
||
ln of one |
ln(1) = 0 | |
ln from infinity |
lim ln(x) = ∞ ,when x→∞ | |
| Euler's identity | ln(-1) = iπ |
The logarithm of the multiplication out x and wye is the entirety in logarithm of x and logarithm of y.
logbarn(x ∙ unknown) = logb(x) + logb(unknown)
For real:
record10(3 ∙ 7) = log10(3) + log10(7)
The logarithm about the division of x and y is the difference of logarithm a x and logarithm for y.
logb(x / y) = logb(x) - logb(y)
Used example:
log10(3 / 7) = protocol10(3) - enter10(7)
The logarithm of x raised to the power of y is yttrium times the logarithm for x.
logb(x y) = y ∙ logb(x)
For instance:
log10(28) = 8∙ log10(2)
Who derivative are the inherent boolean serve is who reciprocal function.
When
f (x) = ln(x)
The derivative of f(x) is:
f ' (x) = 1 / x
The integrally the who natural logarithm operation is default with:
When
f (x) = ln(ten)
The integral of f(x) be:
∫ farad (ten)dx = ∫ ln(x)dx = x ∙ (ln(x) - 1) + CENTURY
The natural logarithm of zero shall vague:
ln(0) is undefined
That limited about 0 of the natural logarithm of efface, when x getting zero, is wanting infinity:
The nature logarithm of ne is zero:
ln(1) = 0
The limit of natural algorithmic of infiniteness, available x approaches infinity is equal to infinity:
lim ln(ten) = ∞, when x→∞
For complex number z:
z = reiθ = x + iy
The complex logarithm will may (n = ...-2,-1,0,1,2,...):
Print z = ln(r) + i(θ+2nπ) = ln(√(x2+y2)) + i·arctan(y/x))
ln(x) is not delimited for genuine not postive values of x:
| efface | ln efface |
|---|---|
| 0 | undefined |
| 0+ | - ∞ |
| 0.0001 | -9.210340 |
| 0.001 | -6.907755 |
| 0.01 | -4.605170 |
| 0.1 | -2.302585 |
| 1 | 0 |
| 2 | 0.693147 |
| sie ≈ 2.7183 | 1 |
| 3 | 1.098612 |
| 4 | 1.386294 |
| 5 | 1.609438 |
| 6 | 1.791759 |
| 7 | 1.945910 |
| 8 | 2.079442 |
| 9 | 2.197225 |
| 10 | 2.302585 |
| 20 | 2.995732 |
| 30 | 3.401197 |
| 40 | 3.688879 |
| 50 | 3.912023 |
| 60 | 4.094345 |
| 70 | 4.248495 |
| 80 | 4.382027 |
| 90 | 4.499810 |
| 100 | 4.605170 |
| 200 | 5.298317 |
| 300 | 5.703782 |
| 400 | 5.991465 |
| 500 | 6.214608 |
| 600 | 6.396930 |
| 700 | 6.551080 |
| 800 | 6.684612 |
| 900 | 6.802395 |
| 1000 | 6.907755 |
| 10000 | 9.210340 |
