SOLUTION: Use the given information and the properties of logarithms to find a requested value. If log base b 2 = 3 and log base b 16 = 4 find log base b 32

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Use the given information and the properties of logarithms to find a requested value. If log base b 2 = 3 and log base b 16 = 4 find log base b 32      Log On


   

Question 419982: Use the given information and the properties of logarithms to find a requested value. If log base b 2 = 3 and log base b 16 = 4 find log base b 32
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
First of all, there is something wrong with this problem:
  • If log%28b%2C+%2816%29%29+=+4 then b%5E4+=+16.
  • If b%5E4+=+16 then b = 2 or -2.
  • Since b is the base of a logarithm it cannot be negative. So b must be 2.
  • If b = 2 then log%28b%2C+%282%29%29+=+log%282%2C+%282%29%29+=+1 not 3!
  • So the equations log%28b%2C+%282%29%29+=+3 and log%28b%2C+%2816%29%29+=+4 are inconsistent. (IOW: It is not possible for both to be true at the same time.

If you posted the problem exactly as it was given to you, then your teacher (or the people who wrote the textbook) made a mistake. If you decide to bring this to your teacher's attention, the please do so respectfully.

If we pretend that there there is no error in the problem itself, then the problem is to rewrite log%28b%2C+%2832%29%29 in terms of the two logarithms your were given. Since 32 = 16*2 we can rewrite log%28b%2C+%2832%29%29 as:
log%28b%2C+%2816%2A2%29%29
Now we can use a property of logarithms, log%28a%2C+%28p%2Aq%29%29+=+log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29, which allows us to expression the log of a product as the sum of the logs of the factors:
log%28b%2C+%2816%29%29+%2B+log%28b%2C+%282%29%29
We have now expressed log%28b%2C+%2832%29%29 in terms of the two logs you were given. We can replace these logs with the values we were given for them:
4 + 3
which simplifies to
7