Exponentials and Logarithms

Exponentials and Logarithms

Characteristics of Exponential Functions

  • Understand that an exponential function has the form y = a^x, where a is a positive real number not equal to 1.
  • Recognise the key features of exponential graphs including the y-intercept at 1 and asymptote at 0 for a > 1, or y-intercept at 1 and asymptote at 0 for 0 < a < 1.
  • Understand that exponential functions increase faster (a > 1) or decrease faster (0 < a < 1) as x increases.
  • Recognise that the base number of an exponential function determines whether the function is increasing or decreasing.

Properties of Exponents

  • Understand Product of powers rule: a^m * a^n = a^(m+n).
  • Recognise Power of a power rule: (a^m)^n = a^(mn).
  • Utilise Quotient of powers rule: a^m / a^n = a^(m-n).
  • Apply Zero exponent rule: a^0 = 1.
  • Understand Negative exponent rule: a^-n = 1/a^n.

Logarithmic Functions

  • Note that the logarithm of a number is the exponent to which another fixed value, the base, must be raised to produce that number.
  • Recognise the logarithm base change rule: logb(a) = logc(a) / logc(b).
  • Understand that logarithmic functions are the inverses of exponential functions.

Properties of Logarithms

  • Understand Product of logarithm rule: logb(mn) = logb(m) + logb(n).
  • Note the Quotient of logarithm rule: logb(m/n) = logb(m) - logb(n).
  • Apply Power of logarithm rule: logb(m^n) = n logb(m).
  • Recognise logarithms base change of base rule: logb(a) = logc(a) / logc(b).

Solving Equations with Logarithms and Exponents

  • Use logarithms to solve exponential equations.
  • Apply laws of logarithms to simplify logarithmic expressions.
  • Understand how to use logarithms to solve logarithmic equations.

Common Pitfalls

  • Avoid common mistakes such as thinking that logb(m/n) = logb(m) / logb(n).
  • Be aware that logarithms do not distribute over addition or subtraction.
  • Beware of the domain of logarithm and exponential functions; they are only defined for positive numbers.

Solving Modelling Problems

  • Understand how to apply exponential growth and decay in real-life applications such as populations, radioactive decay, and interest rates.
  • Know how to use the natural logarithm, ln, in calculations particularly in compound interest problems.
  • Be aware of how to convert between logarithmic and exponential form to solve problems.