How do you solve # 5 times 6^x = 6 times 5^x#?

1 Answer
May 14, 2016

#x=1#

Explanation:

Given,

#5*6^x=6*5^x#

Take the logarithm of both sides since the bases are not the same.

#log(5*6^x)=log(6*5^x)#

Using the logarithmic property, #log_color(purple)b(color(blue)(x)*color(red)(y))=log_color(purple)b(color(blue)(x))+log_color(purple)b(color(red)(y))#, the equation becomes,

#log(5)+log(6^x)=log(6)+log(5^x)#

Group all terms with #x# on one side of the equation.

#log(6^x)-log(5^x)=log(6)-log(5)#

Using the logarithmic property, #log_color(purple)b(color(blue)x^color(red)y)=color(red)y*log_color(purple)b(color(blue)x)#, the equation becomes,

#xlog(6)-xlog(5)=log(6)-log(5)#

Factor out #x# from the left side.

#x(log(6)-log(5))=log(6)-log(5)#

Solve for #x#.

#x=(log(6)-log(5))/(log(6)-log(5))#

#color(green)(|bar(ul(color(white)(a/a)color(black)(x=1)color(white)(a/a)|)))#