Exponentials and Logarithms

Introduction to Exponentials and Logarithms

  • Exponential functions are functions of the form y = ax, a > 0
  • The basic property of an exponential function is that a quantity increases or decreases at a rate proportional to its current value.
  • The logarithmic function is the inverse of an exponential function. It is denoted by y = loga(x)
  • The base ‘a’ of the logarithmic function is the base of the exponential function to which it acts as an inverse.
  • In logarithmic functions, log(a.b) = log(a) + log(b) and log(a/b) = log(a) - log(b). It is essential to grasp these properties.

Important Properties of Exponentials

  • The function y = ax has a horizontal asymptote at y=0. For a > 1, this is an exponential growth, while for 0 < a < 1, it’s an exponential decay.
  • The function is always positive: ax > 0 for all values of x.
  • The function y = ax + b has a horizontal asymptote at y = b. This vertical shift is significant.
  • ax+y = ax.ay and (ax)y = ax.y

Main Properties of Logarithms

  • Inverse relation: loga(ax) = x and aloga(x) = x. These equations express the inverse nature of logarithmic and exponential functions.
  • Change of base rule: logb(a) = logc(a) / logc(b). This rule allows logarithms to change their base to facilitate calculations.
  • Power rule: loga(bn) = n.loga(b). This property allows the manipulation of exponents within logarithmic expressions.
  • loga(1) = 0 regardless of base a, and loga(a) = 1 regardless of base a.

Solving Exponential and Logarithmic Equations

  • To solve exponential equations, like ax = b, rewrite the equation in logarithmic form: x = loga(b).
  • To solve logarithmic equations such as loga(x) = b, convert to exponential form: ab = x.
  • Remember to check the domain of your answers. For example, loga(x) doesn’t exist for x ≤ 0.

Applications of Exponentials and Logarithms

  • Exponential and logarithmic functions are used in many scientific fields, like physics, biology, and computer science.
  • They model phenomena where things grow or decay at a rate proportional to their size, such as populations (growth) or radioactive materials (decay).
  • Logarithmic scales, like the Richter scale, are used for phenomena with huge variations in size.