Chapter 3 - Section 3.4 - Exponential and Logarithmic Equations - Exercise Set - Page 491: 123

$(1)$ $\log_{3}(x-1)=4$ is solved using the definition of a logarithm, (if $ y=\log_{b}x $ then $ x=b^{y}$), so the next step would be $ x-1=3^{4}$ $ x=81-1=80.$ $(2)$ $\log_{3}(x-1)=\log_{3}4$ is solved by applying the one-to-one property of logarithmic functions, (if $\log_{b}M=\log_{b}N,$ then $ M=N $), so the next step is $ x-1=4$ $ x=5$

Given above.