How do you solve # 5^(x - 1) = 3^x#?

1 Answer
Mar 16, 2016

#x~~3.15#

Explanation:

#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)|)))#