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maths:log_exp

a brief summary of logarithms and exponential functions

Introduction

  • 1614, John Napier, publicly propounded the method of logarithms
  • in the 18thC, Leonhard Euler gave the current definition of the 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

Logarithmic functions:

  • loga(x) + loga(y) = loga(xy);
  • log(1) = 0;
  • loga(x) - loga(y) = loga(x/y);
  • ploga(x) = loga(x^p);
  • logb(x)= loga(x)/loga(b);
  • complex numbers in logs:
    • log(cosb + isinb) = ib

Exponential functions:

  • a^m * a^n = a^(m+n);
  • a^m/a^n = a^(m-n);
  • (a^m)^n = a^(mn);
  • (ab)^n = a^n * b^n;
  • a^(-n) = 1/a^n;
  • a^0 = 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.

Hyperbolic functions:

  • sinhx = 0.5(e^x - e^(-x));
  • coshx = 0.5(e^x + e^(-x));
  • tanhx = (e^x - e^(-x))/(e^x + e^(-x));
maths/log_exp.txt · Last modified: 2021/07/24 11:38 by gary1