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1,  Elements of Reason. 1996
1A. Pattern Based Reason  1995
1B. Math Curriculum Notes 1996
2. Three Skills for Algebra  1995
3._Why_Slopes_&_More_Math_1995

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Natural Logarithms and Exponentials

The natural logarithm ln(x) is defined for x > 0.   The exponential function exp(x) is defined for all real x as the inverse of the natural logarithm - the function that gives the value of a given the value of b = ln(a).  For further explanation, see chapter 19 in the online book Why Slopes and More Math.

1.  Fundamental properties of logarithms and exponential functional,
   
   (i) ln(ab) = ln(a) + ln(b) when a and b positive, and 
   
   (ii) exp(x+y) = exp(x) exp(y) for x and y real,
   
   (iii) exp (ln(x)) = x  for x > 0 and ln(exp(x)) = x for all real x.
   
   
2. Properties  (i) and (ii)  cand be used to define and derive the properties of other logarithms and exponentials.

Uniqueness (or 1 to 1) Property:  If a > 0, b> 0 and  ln(a) = ln(b) then a = b.

Proof: (i) the horizontal line y = ln(a) also described by y = ln (b)   intersects the  curve y =ln(x) at only one point. So the function y = ln(x) is one to one. ....

Inversion Properties

• ln(exp(x)) = x for all real x

• exp(ln(x)) = x if x > 0

For each real number a,   x = exp(a) is the unique solution of  a =  ln(x).  Solving the latter equation  is one way to define or compute exp(a). 

Fundamental property of logarithms  ln(ab) = ln(b) +ln(a)    (proof available in calculus)

Fundamental property of exponentials: exp(x₁) · exp(x₂) = exp(x₁+x₂)  This follows from the uniqueness property of logarithms and the fundamental properties of logarithms.

The fundamental property of logarithms implies 

• ln( 1/a) = (-1) ln(a) as  0 = ln(1) = ln ( (1/a) a )

• ln(a ^m) = m ln (a) for all whole numbers and then for all integers. integers.