SOLUTION: log base 3 of 31 So far I got log 3 (15.5)+ log 3 (2)

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Question 412677: log base 3 of 31
So far I got log 3 (15.5)+ log 3 (2)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%283%2C+%2831%29%29
I assume you are supposed to figure out a decimal approximation for this log. (If not, repost it and explain what the problem says for you to do.)

What you did was correct but it did not get you any closer to an answer.

There are two ways to figure out a logarithm. Some logarithms can be found "by hand". For example, log%284%2C+%2816%29%29 represents the exponent for 4 that results in 16. This exponent we know. It is 2. So log%284%2C+%2816%29%29 = 2.

For logarithms where you can't figure out the right exponent you must use your calculator. Your expression log%283%2C+%2831%29%29 represents the exponent on a 3 that results in 31. This cannot be done without a calculator because the power of 3 that results in 31 is not well known.

Unfortunately calculators do not have buttons for base 3 logarithms. In order to use our calculators we must change the base of the logarithm, using the change of base formula, to a base your calculator "knows", like base 10 or base e (aka ln). The change of base formula is log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29. Using this pattern on your expression we get:
log%283%2C+%2831%29%29+=+log%28%2831%29%29%2Flog%28%283%29%29
or
log%283%2C+%2831%29%29+=+ln%2831%29%2Fln%283%29
Believe it or not, both of these equations will give the same result. I will leave it up to you to pick one, use your calculator (or computer calculator program) to find the two logarithms and then divide.

P.S. In response to your comment about being without a calculator...
  • You need a calculator only if you need/want a decimal approximation of the answer. Otherwise
    log%283%2C+%2831%29%29+=+log%28%2831%29%29%2Flog%28%283%29%29
    or
    log%283%2C+%2831%29%29+=+ln%2831%29%2Fln%283%29
    would be the preferred answer because they are exactly correct.
  • You obviously have access to a computer. Most computers have a calculator program on them. If you are using Windows you will probably find it under
    Start -> Programs -> Accessories
    If not, then try
    Start -> Run...
    and type in "calc" (without the quotes) followed by the Enter key.
    Once you start the calculator program, click on the "View" menu and select "Scientific". This should change the display so that it includes more buttons, including log and ln!