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

2 Answers
Mar 10, 2018

Use the law

#log(a) - log(b) = log(a/b)#.

Hence

#log((x+ 1)/(x- 1)) = 1#

#(x + 1)/(x - 1) = 10#

#x +1 = 10(x - 1)#

#x + 1 = 10x - 10#

#11 = 9x#

#x = 11/9#

We now verify that it satisfies the equation. It will as long as #x > 1#, which it is. Hence, we're done.

Hopefully this helps!

Mar 10, 2018

The solution is #x=11/9#.

Explanation:

Use these #log#arithm rules to help simplify the equation:

# log(color(red)x)color(green)-log(color(blue)y)=log(color(green)(color(red)x/color(blue)y)) #

# log(x)=log(y)qquadquad=>qquadquadx=y #

Now here's the actual problem:

# log(color(red)(x+1))color(green)-log(color(blue)(x-1))=1 #

# log(color(green)(color(red)(x+1)/color(blue)(x-1)))=1 #

# log(color(green)(color(red)(x+1)/color(blue)(x-1)))=log(10) #

# color(green)(color(red)(x+1)/color(blue)(x-1))=10 #

# color(red)(x+1)=10(color(blue)(x-1)) #

# color(red)(x+1)=10x-10 #

# color(red)x+11=10x #

# 11=9x #

# 11/9=x #

# x=11/9 #

That's the solution. Hope this helped!