Question

multiple choice 1 point what is the approximate solution to the equation $3^{x - 1}=4^{2x + 5}$? $-4.797$ $-2.354$ $1.262$ $3.875$

Explanation:

Step1: Take natural log on both sides

Given equation \(3^{x - 1}=4^{2x + 5}\). Taking \(\ln\) on both sides, we get \(\ln(3^{x - 1})=\ln(4^{2x + 5})\).
Using the property \(\ln(a^b)=b\ln(a)\), this becomes \((x - 1)\ln(3)=(2x + 5)\ln(4)\).

Step2: Expand both sides

Expand the left - hand side: \(x\ln(3)-\ln(3)\).
Expand the right - hand side: \(2x\ln(4)+5\ln(4)\).

Step3: Rearrange terms to solve for x

Bring all terms with \(x\) to one side and constants to the other side:
\(x\ln(3)-2x\ln(4)=5\ln(4)+\ln(3)\)
Factor out \(x\) from the left - hand side: \(x(\ln(3)-2\ln(4)) = 5\ln(4)+\ln(3)\)

We know that \(\ln(3)\approx1.0986\) and \(\ln(4)\approx1.3863\).
Substitute these values:
\(x(\ 1.0986-2\times1.3863)=5\times1.3863 + 1.0986\)
First, calculate the coefficients:
\(1.0986-2\times1.3863=1.0986 - 2.7726=-1.674\)
\(5\times1.3863+1.0986 = 6.9315+1.0986 = 8.0301\)

Then, solve for \(x\): \(x=\frac{8.0301}{-1.674}\approx - 4.797\)

Answer:

\(-4.797\) (the option corresponding to \(-4.797\))