Log, is this correct?

Question

Hello, wondering if these are correct:

1. log(sub)2 (x+1) - log(sub)2 x = 3
2^log2 (x+1) - 2^log2x = 2^3
(x+1) - x = 8
0 = 7

No solution ?

2. log(sub)4 (x+2) - log(sub)4 (x-3) = 2
4^log4(x+2) - 4^log4(x-3) = 4^2
(x+2) - (x-3) = 16
5 = 16

No solution?

Answer 1

You have to combine the logs before taking them to the power of 2/4.

1. log(sub)2 (x + 1) - log(sub)2 x = 3
log(sub)2 ((x + 1)/x) = 3
2^log2 ((x + 1)/x) = 2^3
1 + 1/x = 2 ^ 3
1/x = 7
x = 1/7

2. log4 (x + 2) - log4 (x - 3) = 2
log4((x + 2)/(x- 3)) = 2
4 ^ log4((x + 2)/(x- 3)) = 4 ^ 2
(x + 2) / (x - 3) = 16
Blegh. algebra.
x = 3 1/3

Answer 2

No, there are solutions for them. Your method are not quite right sir/madam. So to solve this problem you need to know several properties of log. hence:

1. log(sub)2 (x+1) - log(sub)2 x = 3

re-write as:

i) log(sub)2 ((x+1)/ x)= 3 (property of log)

this is same as writing
ii) 2^3 = (x+1)/ x (property of log)

solving this yields
8 = (x+1)/ x
8x = (x +1)
7x = 1
x= 1/7 ..answer ...:)


similarly for #2.
1. log(sub)4 (x+2) - log(sub)4 (x-3) = 2

re-write as:

i) log(sub)4 ((x+2)/( x-3))= 2 (property of log)

this is same as writing
ii) 4^2 = (x+2)/( x-3) (property of log)

solving this yields
16 = (x+2)/ (x -3)
16x - 48 = (x +2)
15x = 50
x= 10/3 ..answer ...:)

hope this helps. by the way you can google to properties of log. :) cheers.

Answer 3

log (x+1) - log x = 3
log(x+1)/x) = 3
(x+1)/x = 2^3 = 8
x+1= 8x
1=7x
1/7= x

log((x+2)/(x-3)) = 2
(x+2)/(x-3) = 2^4 = 16
x+2 = 16x - 48
15x =50
x = 50/15 =10/3

Answer 4

1. log sub2 (x+1) - log sub2 (x) = log sub2 [(x+1)/x] from properties of log

log sub2 [(x+1)/x]=3
(x+1)/x = 8
x+1 = 8x
x=1/7

2. log sub4 [(x+2)/(x-3)] = 2
(x+2)/(x-3) = 16
16x - 48 = x+2
15x=50
x=10/3