SOLUTION: how evaluate log2/3 27/8

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Question 272856: how evaluate log2/3 27/8
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%282%2F3%2C+%2827%2F8%29%29
Here's a couple of ways to evaluate this:
  • Since calculators do not know how to calculate base 2/3 logarithms, we need to change the base to one your calculator "knows" (like base 10 or base e (ln)). There is a formula for chaning the base of a logarithm:
    log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29
    Using this to change your base 2/3 log to a base 10 log we get:
    log%282%2F3%2C+%2827%2F8%29%29+=+log%28%2827%2F8%29%29%2Flog%28%282%2F3%29%29
    We can use our calculators to find the two base 10 logarithms and then divide.
  • A clever, quick and exact answer can be found if we really understand what logarithms are and how exponents work. log%282%2F3%2C+%2827%2F8%29%29 represents the exponent for 2/3 that results in 27/8. In other words, log%282%2F3%2C+%2827%2F8%29%29 is the answer to the question: "What power of 2/3 is 27/8?". We can answer this question if we know our exponents well. 27 is not an obvious power of 2 and 8 is not an obvious power of 3. But, if the fraction was upside down, then 8 is 2%5E3 and 27 is 3%5E3. So if the fraction was 8/27 and not 27/8 then the answer to our question would be 3. The missing piece, then, is what aspect of exponents makes a fraction flip upside down? Answer: Negative exponents mean reciprocals. So "What power of 2/3 is 27/8?". Answer: -3.

Note: The exact, correct answer is -3. If you changed the base and used your calculator you might get -3 or something very close to it like -3.00000001 or -2.99999998. This may happen because your calculator uses rounded-off decimal approximations for:
  • repeating decimals like 2/3
  • most logarithms

So there are three decimal approximations being used by the calculator and this may result in very small errors. So I hope you understood the second solution because it is faster, easier and exact when it can be used.