Question

solve. round your answer to the nearest thousandth. 3 = 9^{2x} - 4 x = \boxed{}

Explanation:

Step1: Isolate the exponential term

Add 4 to both sides of the equation \(3 = 9^{2x}-4\) to get \(9^{2x}=3 + 4\).
\(9^{2x}=7\)

Step2: Rewrite the base

Since \(9 = 3^{2}\), we can rewrite the left - hand side as \((3^{2})^{2x}\). Using the power of a power rule \((a^{m})^{n}=a^{mn}\), we have \((3^{2})^{2x}=3^{4x}\), so the equation becomes \(3^{4x}=7\) (or we can also work with base 9 directly). Taking the logarithm of both sides. Let's take the natural logarithm (we could also use common logarithm).
\(\ln(9^{2x})=\ln(7)\)
Using the logarithm power rule \(\ln(a^{b}) = b\ln(a)\), we get \(2x\ln(9)=\ln(7)\)

Step3: Solve for x

First, we know that \(\ln(9)=\ln(3^{2}) = 2\ln(3)\), but we can also solve for x directly from \(2x\ln(9)=\ln(7)\).
\(x=\frac{\ln(7)}{2\ln(9)}\)
We can also note that \(9 = 3^{2}\), so \(\ln(9)=2\ln(3)\), then \(x=\frac{\ln(7)}{4\ln(3)}\)
Now, calculate the numerical value:
\(\ln(7)\approx1.9459\), \(\ln(9)\approx2.1972\)
\(2\ln(9)\approx4.3944\)
\(x=\frac{1.9459}{4.3944}\approx0.443\)

Answer:

\(0.443\)