Question

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

Answer 1

`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)|)))`