Question

How do you solve `log (2x + 1) - log (x - 2) = 1`?

Answer 1

`x=21/8`

Explanation:

`1`. Use the log property, `log_color(purple)b(color(red)m/color(blue)n)=log_color(purple)b(color(red)m)-log_color(purple)b(color(blue)n)` to simplify the left side of the equation.

`log(2x+1)-log(x-2)=1`

`log((2x+1)/(x-2))=1`

`2`. Use the log property, `log_color(purple)b(color(purple)b^color(orange)x)=color(orange)x`, to rewrite the right side of the equation.

`log((2x+1)/(x-2))=log(10)`

`3`. Since the equation now follows a "`log=log`" situation, where the bases are the same on both sides of the equation, rewrite the equation without the "log" portion.

`(2x+1)/(x-2)=10`

`4`. Solve for `x`.

`2x+1=10(x-2)`

`2x+1=10x-20`

`8x=21`

`color(green)(|bar(ul(color(white)(a/a)x=21/8color(white)(a/a)|)))`