Question

How do you solve `log_[5] (24x + 4) - 2log_[5] (x - 2) = 2`?

Answer 1

`x=4`

Explanation:

`1`. Use the logarithmic property, `log_color(purple)b(color(red)m^color(blue)n)=color(blue)n*log_color(purple)b(color(red)m)`, to rewrite `2log_5(x-2)`.

`log_5(24x+4)-2log_5(x-2)=2`

`log_5(24x+4)-log_5((x-2)^2)=2`

`2`. Use the logarithmic 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_5((24x+4)/(x-2)^2)=2`

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

`log_5((24x+4)/(x-2)^2)=log_5(5^2)`

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

`(24x+4)/(x-2)^2=5^2`

`5`. Isolate for `x`.

`24x+4=25(x-2)^2`

`24x+4=25(x^2-4x+4)`

`24x+4=25x^2-100x+100`

`color(darkorange)25x^2` `color(turquoise)(-124)x` `color(violet)(+96)=0`

`6`. Use the quadratic formula to solve for `x`.

`color(darkorange)(a=25)color(white)(XXXXX)color(turquoise)(b=-124)color(white)(XXXXX)color(violet)(c=96)`

`x=(-b+-sqrt(b^2-4ac))/(2a)`

`x=(-(color(turquoise)(-124))+-sqrt((color(turquoise)(-124))^2-4(color(darkorange)(25))(color(violet)(96))))/(2(color(darkorange)(25)))`

`x=(124+-sqrt(15376-9600))/50`

`x=(124+-sqrt(5776))/50`

`x=(124+-76)/50`

`x=(124+76)/50color(white)(i),color(white)(i)(124-76)/50`

`x=200/50color(white)(i),color(white)(i)48/50`

`color(green)(|bar(ul(color(white)(a/a)x=4color(white)(i),color(white)(i)color(red)cancelcolor(green)(24/25)color(white)(a/a)|)))`

However, if you substitute `x=24/25` back into the original equation, you will find that you will end up taking the logarithm of a `color(red)("negative number")`, which is `color(red)("not possible")`.

For example:

`log_5(24x+4)-2log_5(x-2)=2`

`log_5(24(24/25)+4)-2log_5(24/25-2)=2`

`log_5(676/25)-2log_5(color(red)(-26/25))=2`

For this reason, the correct and only solution to the given logarithmic equation is `x=4`.