a brief summary of logarithms and exponential functions

see also:

• a brief summary of mathematical principles and tools
• History of Mathematics
• mensuration - how to measure area, volume, etc of various shapes
• Algebra
• Calculus
• Circular functions
• Coordinate geometry, parabolae, etc
• Finance equations
• Probability theory

• 1614, John Napier, publicly propounded the method of logarithms

• in the 18thC, Leonhard Euler gave the current definition of the irrational number constant e, the base of the natural logarithm, still known as Euler's number and introduced the use of the exponential function and logarithms in analytic proofs
• e = lim (n ⇒ ∞) (1 + 1/n)ⁿ = 2.171828….
  • ie. a compound interest calculation for 1 time period with an interest rate per time period of 1, and the answer gets very close to the actual value e when you make n = 1,000,000
  • NB. principal + compound interest after t time periods = principal x (1 + r/n)^nt where r is the interest rate per time period and n is the number of times it is compounded during that time period
• e^x is NOT defined as being equal to e multiplied by itself x times as is the case in usual terminology of 2^x although for x as a Real Number, it does equate to this, but does not if x is a complex number
  • INSTEAD it refers to a different infinite polynomial function, often called exp(x) where x can be a real number, complex number, matrix or a quantum operator
    • exp(x) = 1 + x + (x)²/2 + (x)³/6 + ….+ (x)^N/N! = sum( (x)^N/N! ) for N = 0 to however much you want to get to, eg. 100, remembering that for N= 0, 0! = 1 which is the 1st term in our equation, and N=1 gives x, the 2nd term in our equation.
    • ie. exp(0) = e⁰ = 1
    • ie. exp(1) = e¹ = e = 1 + 1 + (1)²/2 + (1)³/6 + ….+ (1)^N/N! = 2.71828…
• e^x is unique in that its differential is also e^x
  • this means the function e^x is unique in that its rate of change (derivative) at any point is equal to its own value at that point!
• exp(a+b) = e^(a+b) = e^a x e^b
  • this means that when x is a Real number, e^x DOES equate to e multiplied by itself x times as:
    • eg. if x = 5 then exp(5) = e⁵ = e^(1+1+1+1+1) = e¹ x e¹ x e¹ x e¹ x e¹ = e x e x e x e x e
• when e is used as the base for a logarithm, it is called the “natural logarithm” and denoted by ln instead of log
• see also complex number maths for the critical use of e in complex number maths

• log_a(x) + log_a(y) = log_a(xy);
• log(1) = 0;
• log_a(x) - log_a(y) = log_a(x/y);
• plog_a(x) = log_a(x^p);
• log_b(x)= log_a(x)/log_a(b);
• log_e(x) = ln(x);
• complex numbers in logs:
  • log_e(cosb + isinb) = ib

• a^m * aⁿ = a^(m+n);
• a^m/aⁿ = a^(m-n);
• (a^m)ⁿ = a^(mn);
• (ab)ⁿ = aⁿ * bⁿ;
• a^-n = 1/aⁿ;
• a⁰ = 1;
• a^1/q, where q is positive integer, a>0,
  • then, = positive qth root of a;
• x^y = Exp(y * Ln(x)); where exp(z) is e to power of z but only when z = Real Number.

• sinh(x) = 0.5(e^x - e^-x);
• cosh(x) = 0.5(e^x + e^-x);
• tanh(x) = (e^x - e^-x)/(e^x + e^-x);