SOLUTION: Question: Using the change of base formula with natural logarithm, find ln x given that log(base 2)x = -1.5. Here is what I have so far... log(base 2)(2x) = log(base 2)(-1.5)

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Question: Using the change of base formula with natural logarithm, find ln x given that log(base 2)x = -1.5. Here is what I have so far... log(base 2)(2x) = log(base 2)(-1.5)       Log On


   

Question 1000465: Question: Using the change of base formula with natural logarithm, find ln x given that log(base 2)x = -1.5.
Here is what I have so far...
log(base 2)(2x) = log(base 2)(-1.5)
Here I took the log of both sides. Log base 2 of 2 becomes 1 and I am left with x on the left of the equation.
x = log(base 2)(-1.5)
I then use change of base and it becomes
x=ln(-1.5)/ln(2) After this point I am not sure if I did it correctly but I suspect this is undefined because the numerator is a natural log of a negative number which is undefined. Can someone tell me if I am missing a step or did i get this correct. It's multiple choice and one of the answers is none of the above but which I am hoping is correct. Please help me understand if I am doing this correct.

Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20855) About Me  (Show Source):
You can put this solution on YOUR website!
log%282%2C%282x%29%29=-1.5......you can write 1.5 as 3%2F2
log%282%2C%282x%29%29=-%283%2F2%29 .......change the base
log%28%282x%29%29%2Flog%28%282%29%29=-%283%2F2%29
log%28%282x%29%29=-%283%2F2%29log%28%282%29%29
log%28%282x%29%29=log%28%282%5E%28-3%2F2%29%29%29
log%28%282x%29%29=log%28%281%2Fsqrt%282%5E3%29%29%29
log%28%282x%29%29=log%28%281%2F2%282%5E%281%2F2%29%29%29%29.....if log same, we have

log%28%28x%29%29=0-log%28%282%29%29-%281%2F2%29log%282%29%29
log%28%28x%29%29=-log%28%28%282%5E%28-3%2F2%29%29%29%29
log%28%28x%29%29=-log%28sqrt%282%5E3%29%29
x=-sqrt%288%29
x=-2.8284271

x=-3=> approximate solution

Answer by MathTherapy(10858) About Me  (Show Source):
You can put this solution on YOUR website!
Question: Using the change of base formula with natural logarithm, find ln x given that log(base 2)x = -1.5.
Here is what I have so far...
log(base 2)(2x) = log(base 2)(-1.5)
Here I took the log of both sides. Log base 2 of 2 becomes 1 and I am left with x on the left of the equation.
x = log(base 2)(-1.5)
I then use change of base and it becomes
x=ln(-1.5)/ln(2) After this point I am not sure if I did it correctly but I suspect this is undefined because the numerator is a natural log of a negative number which is undefined. Can someone tell me if I am missing a step or did i get this correct. It's multiple choice and one of the answers is none of the above but which I am hoping is correct. Please help me understand if I am doing this correct. 
You have 2 different equations above. However, the one I'm solving for you is:

log+%282%2C+%282x%29%29+=+-+1.5

2x+=+2%5E%28-+1.5%29 --------------- Converting to exponential form

ln+%282x%29+=+ln+%282%5E%28-+1.5%29%29 -------- Taking the natural log of both sides

ln+%282%29+%2B+ln+%28x%29+=+ln+%282%5E%28-+1.5%29%29 --- Applying ln+%28ab%29+=+ln+%28a%29%2B+ln+%28b%29

ln+%28x%29+=+ln+%282%5E%28-+1.5%29%29+-+ln+%282%29 --- Subtracting ln 2 from both sides

ln+%28x%29+=+ln+%282%5E%28-+1.5%29%2F2%29

ln+%28x%29+=+ln+%282%5E%28-+1.5+-+1%29%29

highlight_green%28ln+%28x%29+=+ln+%282%5E%28-+2.5%29%29%29 -------- This is what you asked for, so there’s no need to go further

If I’m wrong and you do need x, then: x+=+2%5E%28-2.5%29