Logarithmic or Indicial Equations - SS2 Mathematics Lesson Note
Example 6Ā Solve the equationĀ \(9^{2x + 1} = \frac{81^{x - 2}}{3^{x}}\)
SolutionĀ 3, 9 and 81 are all powers of 3 and can be expressed thus:Ā \(3^{1},3^{2}\)Ā andĀ \(3^{4}\)
\(3^{2(2x + 1)} = \frac{3^{4(x - 2)}}{3^{x}}\ \rightarrow \ 3^{4x + 2} = \frac{3^{4x - 8}}{3^{x}}\ \rightarrow \ 3^{4x + 2} = 3^{4x - 8 - x}\ \rightarrow \ 3^{4x + 2} = 3^{3x - 8}\ \rightarrow 4x + 2 = 3x - 8\)
\(4x - 3x = - 8 - 2\ \rightarrow x = - 10\)
Example 7Ā EvaluateĀ \(\log_{9}3\)
SolutionĀ \(\log_{9}3 = x\)
\(9^{x} = 3\)
\(3^{2x} = 3^{1}\)
\(2x = 1\)
\(x = \frac{1}{2}\)
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