Logarithmic or Indicial Equations - SS2 Mathematics Lesson Note

Example 6 Solve the equation \(9^{2x + 1} = \frac{81^{x - 2}}{3^{x}}\)

Solution 3, 9 and 81 are all powers of 3 and can be expressed thus: \(3^{1},3^{2}\) and \(3^{4}\)

\(3^{2(2x + 1)} = \frac{3^{4(x - 2)}}{3^{x}}\ \rightarrow \ 3^{4x + 2} = \frac{3^{4x - 8}}{3^{x}}\ \rightarrow \ 3^{4x + 2} = 3^{4x - 8 - x}\ \rightarrow \ 3^{4x + 2} = 3^{3x - 8}\ \rightarrow 4x + 2 = 3x - 8\)

\(4x - 3x = - 8 - 2\ \rightarrow x = - 10\)

Example 7 Evaluate \(\log_{9}3\)

Solution \(\log_{9}3 = x\)

\(9^{x} = 3\)

\(3^{2x} = 3^{1}\)

\(2x = 1\)

\(x = \frac{1}{2}\)