Question

solve. round your answer to the nearest thousandth.
$4 = 3^{x - 1}$
$x = \square$

Explanation:

Step1: Take logarithm on both sides

To solve the equation \(4 = 3^{x - 1}\), we can take the natural logarithm (ln) of both sides. This gives us \(\ln(4)=\ln(3^{x - 1})\).

Step2: Use logarithm power rule

Using the power rule of logarithms, \(\ln(a^b)=b\ln(a)\), we can rewrite the right - hand side as \((x - 1)\ln(3)\). So the equation becomes \(\ln(4)=(x - 1)\ln(3)\).

Step3: Solve for \(x\)

First, we divide both sides of the equation by \(\ln(3)\): \(\frac{\ln(4)}{\ln(3)}=x - 1\). Then, we add 1 to both sides to isolate \(x\): \(x = 1+\frac{\ln(4)}{\ln(3)}\).
We know that \(\ln(4)\approx1.3863\) and \(\ln(3)\approx1.0986\). So \(\frac{\ln(4)}{\ln(3)}\approx\frac{1.3863}{1.0986}\approx1.2619\). Then \(x=1 + 1.2619=2.2619\). Rounding to the nearest thousandth, we get \(x\approx2.262\).

Answer:

\(2.262\)