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

1 Answer
Apr 2, 2016

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.