Question

solve. round your answer to the nearest thousandth.
$2^{3x - 2} = 7$
$x = \square$

Explanation:

Step1: Take log of both sides

Take the natural logarithm (ln) of both sides of the equation \(2^{3x - 2}=7\). We get \(\ln(2^{3x - 2})=\ln(7)\).

Step2: Use logarithm power rule

Using the power rule of logarithms \(\ln(a^b)=b\ln(a)\), the left - hand side becomes \((3x - 2)\ln(2)=\ln(7)\).

Step3: Solve for x

First, divide both sides by \(\ln(2)\): \(3x - 2=\frac{\ln(7)}{\ln(2)}\).
Then, add 2 to both sides: \(3x=\frac{\ln(7)}{\ln(2)} + 2\).
Finally, divide both sides by 3: \(x=\frac{\frac{\ln(7)}{\ln(2)}+2}{3}\).
We know that \(\ln(7)\approx1.9459\) and \(\ln(2)\approx0.6931\).
\(\frac{\ln(7)}{\ln(2)}\approx\frac{1.9459}{0.6931}\approx2.8075\).
Then \(\frac{\ln(7)}{\ln(2)} + 2\approx2.8075+2 = 4.8075\).
And \(x=\frac{4.8075}{3}\approx1.6025\approx1.603\) (rounded to the nearest thousandth).

Answer:

\(1.603\)