Answer
$x\approx -5.057$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
3^{x-4}=7^{2x+5}
,$ 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}
\log3^{x-4}=\log7^{2x+5}
.\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-4)\log3=(2x+5)\log7
.\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(\log3)-4(\log3)=2x(\log7)+5(\log7)
\\\\
x\log3-4\log3=2x\log7+5\log7
.\end{array}
Using the properties of equality to isolate the variable, the equation above is equivalent to
\begin{array}{l}\require{cancel}
x\log3-2x\log7=5\log7+4\log3
\\\\
x(\log3-2\log7)=5\log7+4\log3
\\\\
x=\dfrac{5\log7+4\log3}{\log3-2\log7}
\\\\
x\approx -5.057
.\end{array}
