#1#. Since the left and right sides of the equation do not have the same base, start by taking the log of both sides.
#5^(x-1)=3^x#
#log(5^(x-1))=log(3^x)#
#2#. Use the log property, #log_color(purple)b(color(red)m^color(blue)n)=color(blue)n*log_color(purple)b(color(red)m)#, to simplify both sides of the equation.
#(x-1)log5=xlog3#
#3#. Expand the brackets.
#xlog5-log5=xlog3#
#4#. Group all like terms together such that the terms with the variable, #x#, are on the left side and #log5# is on the right side.
#xlog5-xlog3=log5#
#5#. Factor out #x# from the terms on the left side of the equation.
#x(log5-log3)=log5#
#6#. Solve for #x#.
#x=log5/(log5-log3)#
#color(green)(|bar(ul(color(white)(a/a)x~~3.15color(white)(a/a)|)))#
