Question

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

Answer 1

`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)|)))`