How do you solve log_[5] (24x + 4) - 2log_[5] (x - 2) = 2?
1 Answer
Explanation:
log_5(24x+4)-2log_5(x-2)=2
log_5(24x+4)-log_5((x-2)^2)=2
log_5((24x+4)/(x-2)^2)=2
log_5((24x+4)/(x-2)^2)=log_5(5^2)
(24x+4)/(x-2)^2=5^2
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
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
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