Question 1065221: The following logarithm problem is a catastrophe. I did it three times and was hoping for same answer, but nope. Every time I got a different answer.
TRY 1:
log_b( 28 ) - log_b(7X) = log_b( 8 )
log_b( 28/7 )=log_b( 8 )
log_b( 4/x )=log_b( 8 )
log_b( 4 )-log_b( x )-log_b( 8 )=0
log_b( 4 )-log_b( 8 )=log_b( x )
log_b( x )=log_b( 4 )-log_b( 8 )
Now, I am gonna raise b to each side to get rid of log_b
b^(log_b( x ))=b^(log_b( 4 ))-b^(log_b( 8 ))
x=4-8= -4
This solution is incorrect, and I am not sure why. I think I did everything right to the best of my knowledge.
TRY 2:
log_b( 28 )-log_b( 7X )=log_b( 8 )
log_b( 28 )-log_b( 8 )=log_b( 7X )
Now, I am raising b to each side to get rid of log_b
b^(log_b( 28 ))-b^(log_b( 8 ))=b^(log_b( 7X ))
28-8=7x
20/7=x
This solution is also wrong, and I am not sure why, but to me, everything looks fine.
TRY 3 (the one COPIED from the book):
log_b( 4/x )=log_b( 8 )
Now, I am raising b to each side to get rid of log_b
b^log_b( 4/x )=b^(log_b( 8 ))
4/x=8
4=8x
4/8=x
1/2=x
This solution is correct. I understand how it's derived, but I have no clue why the above two solutions are not the same than this one. I expected them to be the same regardless of how I solve the problem.
Thank you for having the courage to answer this long question.
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39620)   (Show Source):
Answer by MathTherapy(10552)  (Show Source):
You can put this solution on YOUR website!
The following logarithm problem is a catastrophe. I did it three times and was hoping for same answer, but nope. Every time I got a different answer.
TRY 1:
log_b( 28 ) - log_b(7X) = log_b( 8 )
log_b( 28/7 )=log_b( 8 )
log_b( 4/x )=log_b( 8 )
log_b( 4 )-log_b( x )-log_b( 8 )=0
log_b( 4 )-log_b( 8 )=log_b( x )
log_b( x )=log_b( 4 )-log_b( 8 )
Now, I am gonna raise b to each side to get rid of log_b
b^(log_b( x ))=b^(log_b( 4 ))-b^(log_b( 8 ))
x=4-8= -4
This solution is incorrect, and I am not sure why. I think I did everything right to the best of my knowledge.
TRY 2:
log_b( 28 )-log_b( 7X )=log_b( 8 )
log_b( 28 )-log_b( 8 )=log_b( 7X )
Now, I am raising b to each side to get rid of log_b
b^(log_b( 28 ))-b^(log_b( 8 ))=b^(log_b( 7X ))
28-8=7x
20/7=x
This solution is also wrong, and I am not sure why, but to me, everything looks fine.
TRY 3 (the one COPIED from the book):
log_b( 4/x )=log_b( 8 )
Now, I am raising b to each side to get rid of log_b
b^log_b( 4/x )=b^(log_b( 8 ))
4/x=8
4=8x
4/8=x
1/2=x
This solution is correct. I understand how it's derived, but I have no clue why the above two solutions are not the same than this one. I expected them to be the same regardless of how I solve the problem.
Thank you for having the courage to answer this long question.
The following logarithm problem is a catastrophe. I did it three times and was hoping for same answer, but nope. Every time I got a different answer.
TRY 1:
log_b( 28 ) - log_b(7X) = log_b( 8 )
log_b( 28/7 )=log_b( 8 ) This should be 
log_b( 4/x )=log_b( 8 ) You are correct up to here
log_b( 4 )-log_b( x )-log_b( 8 )=0
log_b( 4 )-log_b( 8 )=log_b( x )
log_b( x )=log_b( 4 )-log_b( 8 )
Now, I am gonna raise b to each side to get rid of log_b
b^(log_b( x ))=b^(log_b( 4 ))-b^(log_b( 8 ))
x=4-8= -4
log_b( 4/x )=log_b( 8 ) Picking up where you were last correct
 ------  means that: 
Continue solving for x and you should get: 
TRY 2:
log_b( 28 )-log_b( 7X )=log_b( 8 )
log_b( 28 )-log_b( 8 )=log_b( 7X ) You are correct up to here
Now, I am raising b to each side to get rid of log_b
b^(log_b( 28 ))-b^(log_b( 8 ))=b^(log_b( 7X ))
28-8=7x
20/7=x
log_b( 28 )-log_b( 8 )=log_b( 7X ) Picking up where you were last correct
 ------ AGAIN, means that:
 ------- Reducing fraction on left side
14x = 7 ------ Cross-multiplying
There you go....same answer!
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