| Author |
 Topic  |
|
|
alexdc
Junior Member
USA
5 Posts |
 Posted - 07/30/2011 : 14:28:20
|
I'm not understanding where I'm going wrong on this one, but here it is:
find x when
log (sub 2) x = log (sub 3) x + 1
Answer:
x = 2^( 1/(1-log (sub2) 3) ) or x = 3^(1/(log (sub 2) 3 - 1)
My work:
start by dividing each side by log (sub 2) x to get:
1 = ( (log (sub 3) x) / (log (sub 2) x) ) + ( 1 / (log (sub 2) x) )
then, use the identity:
( (log (sub a) y) / (log (sub b) y) ) = log (sub a) b
A bit of algebra, and this identity is also:
1 / (log (sub b) y) = (log (sub a) b) * (log (sub a) y)
Applying this identity to the problem, we have:
y = x
a = 3
b = 2
So that:
( (log (sub 3) x) / (log (sub 2) x) ) = log (sub 3) 2
and:
( 1 / (log (sub 2) x) ) = (log (sub 3) 2) * (log (sub 3) x)
so our equation is now:
1 = ( log (sub 3) 2 ) + ( (log (sub 3) 2) * (log (sub 3) x) )
Divide both sides by ( log (sub 3) 2 ) to get:
1 / ( log (sub 3) 2 ) = 1 + ( log (sub 3) x )
subtract both sides by 1 to get:
( 1 / ( log (sub 3) 2 ) ) - 1 = ( log (sub 3) x )
we know that log (sub a) b = c is the same as b = a ^ c, so:
x = 3 ^ ( ( 1 / ( log (sub 3) 2 ) ) - 1 )
which is not one of the answers. Similarly, by starting to solve this by dividing the whole equation ( log (sub 3) x ), I get the answer:
2^( 1/(1-log (sub3) 2) )
I'm not seeing what I'm doing wrong here, but my answers don't check, and answers in the book do.
Any help is appreciated! Thank you. |
 |
|
|
royhaas
Moderator
USA
3039 Posts |
Posted - 08/01/2011 : 07:55:01
|
Start by exponentiating each side. For example, if
u = v, then 2^u = 2^v. |
|
|
|
alexdc
Junior Member
USA
5 Posts |
Posted - 08/01/2011 : 16:39:51
|
I think I understand what you are saying, but I'm not seeing the next step. I take it the idea would be to make each side as a power of 2 or 3, and get either (making both sides a power of 2):
x = 2^( log (sub 3) x + 1 )
simplified:
x = 2^( log (sub 3) x ) * (2^1)
simplified:
x = 2^( log (sub 3) x ) * 2
or (making both sides a power of 3):
3^( log (sub 2) x ) = 3^( log (sub 3) x + 1 )
simplified:
3^( log (sub 2) x ) = 3^( log (sub 3) x ) * (3^1)
simplified:
3^( log (sub 2) x ) = 3x
However, in both cases, I'm not sure what to do after the above listed steps. I have a feeling I'm missing something incredibly obvious here. |
Edited by - alexdc on 08/01/2011 16:40:39 |
|
|
|
royhaas
Moderator
USA
3039 Posts |
Posted - 08/02/2011 : 07:28:22
|
| Write the base 2 log in terms of the base 3 log, or vice versa. |
|
|
|
Subhotosh Khan
Advanced Member
USA
9114 Posts |
Posted - 08/02/2011 : 15:25:13
|
quote:
Originally posted by alexdc
I'm not understanding where I'm going wrong on this one, but here it is:
find x when
log (sub 2) x = log (sub 3) x + 1
log3x = log32 * log2x
Answer:
x = 2^( 1/(1-log (sub2) 3) ) or x = 3^(1/(log (sub 2) 3 - 1)
My work:
start by dividing each side by log (sub 2) x to get:
1 = ( (log (sub 3) x) / (log (sub 2) x) ) + ( 1 / (log (sub 2) x) )
then, use the identity:
( (log (sub a) y) / (log (sub b) y) ) = log (sub a) b
A bit of algebra, and this identity is also:
1 / (log (sub b) y) = (log (sub a) b) * (log (sub a) y)
Applying this identity to the problem, we have:
y = x
a = 3
b = 2
So that:
( (log (sub 3) x) / (log (sub 2) x) ) = log (sub 3) 2
and:
( 1 / (log (sub 2) x) ) = (log (sub 3) 2) * (log (sub 3) x)
so our equation is now:
1 = ( log (sub 3) 2 ) + ( (log (sub 3) 2) * (log (sub 3) x) )
Divide both sides by ( log (sub 3) 2 ) to get:
1 / ( log (sub 3) 2 ) = 1 + ( log (sub 3) x )
subtract both sides by 1 to get:
( 1 / ( log (sub 3) 2 ) ) - 1 = ( log (sub 3) x )
we know that log (sub a) b = c is the same as b = a ^ c, so:
x = 3 ^ ( ( 1 / ( log (sub 3) 2 ) ) - 1 )
which is not one of the answers. Similarly, by starting to solve this by dividing the whole equation ( log (sub 3) x ), I get the answer:
2^( 1/(1-log (sub3) 2) )
I'm not seeing what I'm doing wrong here, but my answers don't check, and answers in the book do.
Any help is appreciated! Thank you.
|
|
|
|
alexdc
Junior Member
USA
5 Posts |
Posted - 08/02/2011 : 16:22:18
|
Just noticed that. I screwed up basic algebra, that was my mistake. This:
( (log (sub a) y) / (log (sub b) y) ) = log (sub a) b
DOES NOT equal this:
1 / (log (sub b) y) = (log (sub a) b) * (log (sub a) y)
After ANY amount of algebra. I multiplied both sides by (log (sub a) y), instead of dividing. Stupid mistake, and the answer is correct after using the proper identity.
Thanks for the help! |
|
|
| |
Topic |
|
Interactive Math Goodies Software
find value of x in log equation
