Answer
$x\approx-1.710$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
4^{x-1}=3^{2x}
,$ take the logarithm of both sides. Then use the properties of logarithms and the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Taking the logarithm of both sides, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\log4^{x-1}=\log3^{2x}
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(x-1)\log4=2x\log3
.\end{array}
Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
x(\log4)-1(\log4)=2x\log3
\\\\
x\log4-\log4=2x\log3
.\end{array}
Using the properties of equality to isolate the variable, the equation above is equivalent to
\begin{array}{l}\require{cancel}
x\log4-2x\log3=\log4
\\\\
x(\log4-2\log3)=\log4
\\\\
x=\dfrac{\log4}{\log4-2\log3}
\\\\
x\approx-1.710
.\end{array}
