Answer: \(x=\frac{8+\ln(7/4)}{3}=2.85\) to 3 significant figures.

Answer: \(x=\frac{8+\ln(7/4)}{3}=2.85\) to 3 significant figures.

Start with

\(4e^{2x-3}=7e^{5-x}.\)

Divide both sides by \(4e^{5-x}\):

\(e^{(2x-3)-(5-x)}=\frac74.\)

The exponent simplifies to \(3x-8\), so

\(e^{3x-8}=\frac74.\)

Taking natural logarithms gives

\(3x-8=\ln\frac74.\)

Therefore

\(x=\frac{8+ \ln(7/4)}{3}.\)

Numerically,

\(x=2.85\)

to 3 significant figures.