\(\int \limits_{\ln 2}^x \frac1{e^x - 1}dx = \ln\frac 32\)
\(\Rightarrow \int\limits_{\ln 2}^x \frac{e^{-x}}{1-e^{-x}} dx = \ln\frac 32\)
\(\Rightarrow \log 1 .{e^{{-x}^x}}_{\ln _2} = \ln \frac 32\)
\(\Rightarrow \log 1 - e^{-x} - \log \frac12 = \ln \frac 32\)
\(\Rightarrow \frac{2e^x-1}{e^x} = \frac 32\)
\(\Rightarrow x = \ln 4\)
