Solving Exponential Equations

This guide covers common secondary exponential equations. A multi-term equation permits polynomial substitution only when its powers fit that pattern. Use positive bases different from 1; take logarithms only of positive expressions, and reject substituted values \(u\le0\).

• If both the left and right hand side of the exponential equation have one term
  1. If both sides of the equation can be prime factorised and expressed as the same base. The exponents are equal
     \(9^x = 27^2\)
     \(3^{2x} = 3^6 \)
     Then \(2x = 6 \rightarrow x = 3\)
  2. If the left and right hand side cannot be expressed as the same base, express the equation in logarithmic form
     \(2^x = 7\)
     \(x\lg{2} = \lg{7}\)
     \(x = \frac{\lg{7}}{\lg{2}} \)
• For a multi-term equation whose powers fit a polynomial pattern, make a suitable positive exponential substitution.
  \( \left(2^x \right)^2 - 4 \left(2^x \right) + 3 = 0\)
  Let \(2^x = u\)
  \(u^2 - 4u + 3 = 0\)
  \( (u - 3)(u - 1) = 0\)
  \(u = 3\) or \(u = 1\)
  \(2^x = 3\) or \(2^x = 1\)
  \(x = \frac{\lg{3}}{\lg{2}}\) or \(x = 0\)