Answer:
[tex] \rm x = -\dfrac{7}{2} [/tex]
Step-by-step explanation:
[tex] \rm Solve \: for \: x \: over \: the \: real \: numbers: \\ \rm \longrightarrow 3^{4 x + 5} = \bigg( \dfrac{1}{27} \bigg)^{2 x + 10} \\ \\ \rm \bigg( \dfrac{1}{27} \bigg)^{2 x + 10}= 27^{-2 x - 10}: \\ \rm \longrightarrow 3^{4 x + 5} = 27^{-2 x - 10} \\ \\ \rm Take \: the \: natural \: logarithm \: of \: both \\ \rm sides \: and \: use \: the \: identity \: log \: a^b = b \: log \: a: \\ \rm \longrightarrow log \: 3^ {4 x + 5} = log \: 27 ^{ -2 x - 10} \\ \\ \rm \longrightarrow log \: 3^ {4 x + 5} = log \: 3 ^{3 (-2 x - 10)} \\ \\ \rm \longrightarrow (4x + 5)log \: 3 = 3( - 2x - 10)log \: 3 \\ \\ \rm Divide \: both \: sides \: by \: log \: 3: \\ \rm \longrightarrow 4 x + 5 = 3 (-2 x - 10) \\ \\ \rm Expand \: out \: terms \: of \: the \: right \: hand \: side: \\ \rm \longrightarrow 4 x + 5 = -6 x - 30 \\ \\ \rm Subtract \: 5 - 6 x \: from \: both \: sides: \\ \rm \longrightarrow 10 x = -35 \\ \rm Divide \: both \: sides \: by \: 10: \\ \\ \rm \longrightarrow
x = - \dfrac{7}{2} [/tex]
