document.write( "Question 814157: To four decimal places, the values of log2 and log3 are
\n" ); document.write( "Log2 = 0.3010 and log3 = 0.4771
\n" ); document.write( "Evaluate log (1/9). \n" ); document.write( "

Algebra.Com's Answer #490176 by jsmallt9(3758) $\"\"$ $\"About$

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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):

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 $\"log%28a%2C+%28a%29%29+=+1\"$ . 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.
Use properties of logarithms to rewrite the expression in terms of log's of 2's, 3's and/or 10's.
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 $\"1%2F9+=+3%5E%28-2%29\"$ . So
\n" ); document.write( " $\"log%28%281%2F9%29%29\"$
\n" ); document.write( "can be rewritten as:
\n" ); document.write( " $\"log%28%283%5E%28-2%29%29%29\"$
\n" ); document.write( "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, $\"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\"$
\n" ); document.write( " $\"%28-2%29log%28%283%29%29\"$
\n" ); document.write( "And finally we substitute in the known value for log(3):
\n" ); document.write( " $\"%28-2%29%2A+0.4771\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( "-0.9542 \n" ); document.write( "

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