type the correct answer in the box. use numerals instead of words. if necessary, use / for the fraction bar…

Explanation:

Step1: Isolate the exponential term

Divide both sides of the equation $1.5(4)^{2x}=12$ by $1.5$. We get $(4)^{2x}=\frac{12}{1.5}=8$.

Step2: Take the logarithm of both sides

Take the natural - logarithm of both sides: $\ln(4^{2x})=\ln(8)$. Using the property of logarithms $\ln(a^{b}) = b\ln(a)$, we have $2x\ln(4)=\ln(8)$.

Step3: Solve for x

First, note that $\ln(4)=\ln(2^{2}) = 2\ln(2)$ and $\ln(8)=\ln(2^{3}) = 3\ln(2)$. So the equation $2x\ln(4)=\ln(8)$ becomes $2x\times2\ln(2)=3\ln(2)$. Since $\ln(2)\neq0$, we can divide both sides by $2\ln(2)$. We get $2x=\frac{3}{2}$, then $x = \frac{3}{4}=0.75$.

Answer:

$0.75$