document.write( "Question 594466: Can you please try to show your steps on how you got the following answers?\r
\n" ); document.write( "\n" ); document.write( "Given log[a](5)=2.3 and log[a](3)=1.6, fill in the table below with the appropriate values.\r
\n" ); document.write( "\n" ); document.write( "x 15 ; 9 ; 5/3 ; 5a ; (3)/(a^2)\r
\n" ); document.write( "\n" ); document.write( "log[a](x) \n" ); document.write( "
Algebra.Com's Answer #376886 by jsmallt9(3759)\"\" \"About 
You can put this solution on YOUR website!
Given:
\n" ); document.write( "\"log%28a%2C+%285%29%29=2.3\"
\n" ); document.write( "\"log%28a%2C+%283%29%29=1.6\"
\n" ); document.write( "Plus,
\n" ); document.write( "\"log%28a%2C+%28a%29%29+=+1\" (This is always true, no matter what \"a\" is, so it does not need to be told this.)
\n" ); document.write( "you have been asked to find various other base a logarithms. The \"trick\" to these is to use algebra and/or properties of logarithms to rewrite the desired logs in terms of the logs you already know.

\n" ); document.write( "So for
\n" ); document.write( "\"log%28a%2C+%2815%29%29\"
\n" ); document.write( "we want to rewrite the 15 in terms of 5's, 3's and/or a's. I hope that you can see that 15 is 5*3. Replacing the 15 with 5*3 we get:
\n" ); document.write( "\"log%28a%2C+%285%2A3%29%29\"
\n" ); document.write( "Now we use a property of logarithms for logs of a product, \"log%28x%2C+%28p%2Aq%29%29+=+log%28x%2C+%28p%29%29+%2B+log%28x%2C+%28q%29%29\", we can separate the 5 and 3:
\n" ); document.write( "\"log%28a%2C+%285%29%29+%2B+log%28a%2C+%283%29%29\"
\n" ); document.write( "Now that we have the log of 15 expressed in terms of logs of 5 and 3. We can now use the given values:
\n" ); document.write( "2.3 + 1.6
\n" ); document.write( "which simplifies to 3.9. So \"log%28a%2C+%2815%29%29+=+3.9\"

\n" ); document.write( "\"log%28a%2C+%289%29%29\"
\n" ); document.write( "For 9 we could use either 3*3 or \"3%5E2\". I'll use the later one so you can see another property in use:
\n" ); document.write( "\"log%28a%2C+%283%5E2%29%29\"
\n" ); document.write( "Using a property for logs of a power, \"log%28x%2C+%28p%5Eq%29%29+=+q%2Alog%28x%2C+%28q%29%29\" we can separate the exponent from the 3:
\n" ); document.write( "\"2%2Alog%28a%2C+%283%29%29\"
\n" ); document.write( "Replacing the log with its given value we get:
\n" ); document.write( "2 * 1.6
\n" ); document.write( "which simplifies to
\n" ); document.write( "3.2
\n" ); document.write( "So \"log%28a%2C+%289%29%29+=+3.2\"

\n" ); document.write( "\"log%28a%2C+%285%2F3%29%29\"
\n" ); document.write( "5/3 is already expressed in terms of 5's and 3's. Using a property for logs of quotients, \"log%28x%2C+%28p%2Fq%29%29+=+log%28x%2C+%28p%29%29+-+log%28x%2C+%28q%29%29\" we get:
\n" ); document.write( "\"log%28a%2C+%285%29%29+-+log%28a%2C+%283%29%29\"
\n" ); document.write( "Replacing the logs with their given values we get:
\n" ); document.write( "2.3 - 1.6
\n" ); document.write( "which simplifies to
\n" ); document.write( "0.7
\n" ); document.write( "So \"log%28a%2C+%285%2F3%29%29+=+0.7\"

\n" ); document.write( "\"log%28a%2C+%283%2Fa%5E2%29%29\"
\n" ); document.write( "First we'll use the property for quotients:
\n" ); document.write( "\"log%28a%2C+%283%29%29+-+log%28a%2C+%28a%5E2%29%29\"
\n" ); document.write( "and then the property for powers (on the second log):
\n" ); document.write( "\"log%28a%2C+%283%29%29+-+2%2Alog%28a%2C+%28a%29%29\"
\n" ); document.write( "Now we can replace the logs with their known values.
\n" ); document.write( "1.6 - 2*1
\n" ); document.write( "which simplifies
\n" ); document.write( "1.6 - 2
\n" ); document.write( "-0.4
\n" ); document.write( "So \"log%28a%2C+%283%2Fa%5E2%29%29+=+-0.4\" \n" ); document.write( "
\n" );