List of 11 Natural Log Rules

[kumarrakesh]

Best 11 Natural Log Rules, Explain it.

 

[niyati]

No.	Rule	Formula	Description
1	Definition	y=ln⁡x  ⟺  ey=xy = \ln x \iff e^y = x	ln is the inverse of the exponential function
2	Domain	ln⁡x\ln x defined for x>0x > 0	Only positive arguments allowed
3	ln 1	ln⁡1=0\ln 1 = 0	Because e0=1e^0 = 1
4	ln e	ln⁡e=1\ln e = 1	Because e1=ee^1 = e
5	Product Rule	ln⁡(ab)=ln⁡a+ln⁡b\ln(ab) = \ln a + \ln b	Splits log of a product into sum of logs
6	Quotient Rule	ln⁡ ⁣(ab)=ln⁡a−ln⁡b\ln\!\bigl(\tfrac{a}{b}\bigr) = \ln a - \ln b	Splits log of a quotient into difference of logs
7	Power Rule	ln⁡(ar)=r ln⁡a\ln(a^r) = r\,\ln a	Brings exponent out front
8	Change of Base (to ln)	log⁡ba=ln⁡aln⁡b\log_b a = \tfrac{\ln a}{\ln b}	Converts any log to natural log
9	Inverse Property	ln⁡(ef(x))=f(x)\ln(e^{f(x)}) = f(x)	ln and exp cancel each other
10	Derivative	ddxln⁡x=1x\frac{d}{dx}\ln x = \tfrac{1}{x}	Rate of change of ln x
11	Integral	∫ln⁡x dx=xln⁡x−x+C\int \ln x\,dx = x\ln x - x + C	Antiderivative of ln x

 

[robbin]

Here’s a list of 11 natural log (ln) rules in simple form:

ln(1) = 0

ln(e) = 1

ln(e^x) = x

e^(ln x) = x

ln(ab) = ln(a) + ln(b)

ln(a/b) = ln(a) − ln(b)

ln(a^b) = b × ln(a)

ln(√x) = (1/2)ln(x)

ln(x⁻¹) = −ln(x)

ln|x| = ln(x) for x > 0

d/dx [ln(x)] = 1/x (derivative rule)

 

[stellapotter]

Here are 11 natural logarithm (ln) rules:
ln(1) = 0: log of 1 is always 0.

1. ln(e) = 1: natural log base e.
2. ln(ab) = ln(a) + ln(b): log of product.
3. ln(a/b) = ln(a) - ln(b): log of quotient.
4. ln(aⁿ) = n·ln(a): log of power.
5. ln(√a) = ½·ln(a): root as exponent.
6. e^(ln(x)) = x: inverse function.
7. ln(e^x) = x: ln and e cancel.
8. ln|a| = ln(a) (for a > 0): absolute value keeps it defined.
9. d/dx[ln(x)] = 1/x: derivative of ln.
10. ∫ln(x) dx = x·ln(x) - x + C: integral formula.