Question

How do you solve `16^(x - 4) = 3^(3 - x)`?

Answer 1

`x~~3.72`

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.

`16^(x-4)=3^(3-x)`

`log(16^(x-4))=log(3^(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-4)log16=(3-x)log3`

`3`. Expand the brackets.

`xlog16-4log16=3log3-xlog3`

`4`. Group all like terms together such that the terms with the variable, `x`, are on the left and the ones without on the right.

`xlog16+xlog3=3log3+4log16`

`5`. Factor out `x` from the terms on the left side of the equation.

`x(log16+log3)=3log3+4log16`

`6`. Solve for `x`.

`x=(3log3+4log16)/(log16+log3)`

`color(green)(|bar(ul(color(white)(a/a)x~~3.72color(white)(a/a)|)))`