Question

type the correct answer in the box. use numerals instead of words. if necessary, use / for the fraction bar. what value of x satisfies this equation? $1.5(4)^{2x} = 12$ round your answer to the nearest hundredth. the value of x is \boxed{}.

Explanation:

Step1: Isolate the exponential term

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

Step2: Take logarithms on both sides

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

Step3: Solve for x

First, divide both sides by \(2\ln(4)\): \(x=\frac{\ln(8)}{2\ln(4)}\).
Calculate the values: \(\ln(8)\approx2.0794\), \(\ln(4)\approx1.3863\).
Substitute these values: \(x = \frac{2.0794}{2\times1.3863}=\frac{2.0794}{2.7726}\approx0.75\).

Answer:

\(0.75\)