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

Explanation:

Step1: Isolate the exponential term

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

Step2: Express both sides with the same base

We know that \(4 = 2^2\) and \(8=2^3\). Substitute these into the equation:
\((2^2)^{2x}=2^3\)
Using the exponent rule \((a^m)^n=a^{mn}\), we have \(2^{4x}=2^3\).

Step3: Solve for \(x\)

Since the bases are the same, we can set the exponents equal to each other:
\(4x = 3\)
Divide both sides by \(4\):
\(x=\frac{3}{4}= 0.75\) (Wait, let's check again. Wait, maybe I made a mistake in base conversion. Let's try another way.

Wait, going back to step 1: \(4^{2x}=8\). Let's take logarithms. Take the natural logarithm of both sides:
\(\ln(4^{2x})=\ln(8)\)
Using the logarithm power rule \(\ln(a^b)=b\ln(a)\), we get \(2x\ln(4)=\ln(8)\)
Then \(2x=\frac{\ln(8)}{\ln(4)}\)
Calculate \(\frac{\ln(8)}{\ln(4)}=\frac{3\ln(2)}{2\ln(2)}=\frac{3}{2} = 1.5\) (since \(\ln(8)=\ln(2^3)=3\ln(2)\) and \(\ln(4)=\ln(2^2)=2\ln(2)\))
Then \(2x = 1.5\)
So \(x=\frac{1.5}{2}=0.75\)? Wait, no, wait when we did \(4^{2x}=8\), let's check with original equation. Let's plug \(x = 0.75\) into \(1.5(4)^{2x}\):

\(2x=1.5\), \(4^{1.5}=4^{\frac{3}{2}}=\sqrt{4^3}=\sqrt{64} = 8\), then \(1.5\times8 = 12\), which matches. Wait, but earlier when I did base 2, I had a mistake. Wait, \(4^{2x}=8\), \(4^{2x}=2^{4x}\), \(8 = 2^3\), so \(2^{4x}=2^3\) implies \(4x = 3\) so \(x=\frac{3}{4}=0.75\). Yes, that's correct. Wait, but let's do it with common logarithm.

Alternative way:
From \(4^{2x}=8\), take log base 10:
\(2x\log(4)=\log(8)\)
\(2x=\frac{\log(8)}{\log(4)}\)
\(\log(8)\approx0.9031\), \(\log(4)\approx0.6021\)
\(\frac{0.9031}{0.6021}\approx1.5\)
\(2x = 1.5\)
\(x = 0.75\)

Wait, but let's check again. Wait, the original equation: \(1.5\times4^{2x}=12\). If \(x = 0.75\), \(2x = 1.5\), \(4^{1.5}=8\), \(1.5\times8 = 12\), which is correct. So the value of \(x\) is \(0.75\).

Wait, but maybe I made a mistake in the first base conversion. Let's see: \(4^{2x}=8\), \(4^{2x}=2^{4x}\), \(8 = 2^3\), so \(4x = 3\), so \(x=\frac{3}{4}=0.75\). Yes, that's correct.

Answer:

\(0.75\)