SOLUTION: To four decimal places, the values of log2 and log3 are Log2 = 0.3010 and log3 = 0.4771 Evaluate log (1/9).

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Question 814157: To four decimal places, the values of log2 and log3 are
Log2 = 0.3010 and log3 = 0.4771
Evaluate log (1/9).
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
Here's a procedure for solving these types of problems where they give you the value of some logarithms and ask you to find the value of another log (without using a calculator):
  1. Rewrite the argument of the desired log as a product, quotient and/or powers of the numbers whose logs you know. Note: In addition to the logs you're given you should also know that . So, since we're using base 10 logs in this problem, we should know that log(10) = 1 without it being given to you. So in this problem we start knowing the base 10 logs of 2, 3 and 10.
  2. Use properties of logarithms to rewrite the expression in terms of log's of 2's, 3's and/or 10's.
  3. Substitute in the known values for these logs and simplify.
So for the first step we will want to rewrite 1/9 as a product, quotient and/or power of 2's, 3's and/or 10's. I hope it doesn't take long for you to see that 9 is a power of 3. But what about 1/9? If you know your negative exponents then you will see that . So

can be rewritten as:

Now we need to rewrite this so that it is in terms of log(3). So the -2 has to move. For this we use a property of logarithms,

And finally we substitute in the known value for log(3):

which simplifies to:
-0.9542
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