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)|)))