Natural Logarithms

Graph of a Natural Logarithms


Definition of Natural Logarithm

When

e ^y = x

Then base e logarithm of x is

ln(x) = log_e(x) = y

The e constant or Euler's number is:

e ≈ 2.71828183

Common Logarithms use the Base 10
Natural 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

	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