Solving Exponential Equations

Solving Exponential Equations

TGM Original Questions
  • If both the left and right hand side of the exponential equation 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
      \(2^x = 7\)
      \(x\lg{2} = \lg{7}\)
      \(x = \frac{\lg{7}}{\lg{2}} \)
  • Otherwise, if the exponential equation contains more than one term make a suitable substitution to simplify the problem to solving a polynomial
    \( 3\left(2^x \right)^2 - 4 \left(2^x \right) + 1 = 0\)
    Let \(2^x = u\)
    \(3u^2 - 4u + 1 = 0\)
    \( (u - 3)(u - 1) = 0\)
    \(u = 3\) or \(u = 1\)
    \(2^x = 3\) or \(2^x = 1\)
    \(x = \frac{\lg{3}}{\lg{2}}\) or \(x = 0\)
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