Question

${5}^{x - 2} \times {3}^{2x - 3} = 135 \\ find \: value \: of \: x$
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Answer 1

$HEYA \: \\ \\ 5 {}^{(x - 2)} \times 3 {}^{(2x - 3)} = 5 \times 3 {}^{3} \\ \\ 5 {}^{(x - 2)} \times 5 {}^{ - 1} = 3 {}^{3} \times 3 {}^{ - (2x - 3)} \\ \\ 5 {}^{(x - 3)} = 3 {}^{(6 - 2x)} \\ \\ taking \: log \: on \: both \: sides \: we \: have \\ \\ log5 {}^{(x - 3)} = log {3}^{(6 - 2x)} \\ \\ (x - 3) log(5) = (6 - 2x) log(3) \\ becoz \: \: log(x {}^{n} ) = n log(x) \\ \\ x log(5 ) - 3 log(5) = 6 log(3) - 2x log(3) \\ \\ x log(5) + 2x log(3) = 3 log(5) + 6 log(3) \\ \\ x = 3 log(5) + 6 log(3) \\ \: \: \: \: \: \: \: - - - - - - - - \\ \: \: \: \: \: log(5 ) + 2 log(3) \\ \\ x = 3( log(5) + 2log(6) ) \\ \: \: \: \: \: \: \: \: - - - - - - - - - \\ \: \: \: \: \: \: \: ( log(5) + 2 log(3) ) \\ \\ \\ x = 3$

Answer 2

${5}^{x - 2} \times {3}^{2x - 3} = 135 \\ = > {5}^{x - 2} \times {2}^{2x - 3} = 5 \times {3}^{3} \\ => x - 2 = 1 \: and \: 2x - 3 = 3 \\ => x = 3$