document.write( "Question 418929: Evaluate the expression
\n" ); document.write( "3^log3^8 \n" ); document.write( "

Algebra.Com's Answer #293159 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm guessing that the expression is
\n" ); document.write( " $\"3%5Elog%283%2C+%288%29%29\"$ which means 3 to the base 3 log of 8
\n" ); document.write( "What you posted means 3 to the base 10 log of $\"3%5E8\"$ which is not the same as 3 to the base 3 log of 8.

\n" ); document.write( "Logarithms of bases other than 10 or e are difficult to post because the base is written like a subscript. One way to post your expression would be
\n" ); document.write( "3^(base 3 log of 8)
\n" ); document.write( "Another way would be to post it the way I am getting the expressions to look like they do. Click on the \"Show source\" link just above this solution and you can see what I typed.

\n" ); document.write( "If I am right about your expression then the answer is very easy once you understand what a logarithm is. The idea behind logarithms is this: You can take any positive number except 1 and, if youraise it to the right power, get any positive number. Some of these powers are well-known. For example:
\n" ); document.write( "You can raise 4 to the 2nd power to get 16
\n" ); document.write( "You can raise 4 to the 1/2 power to get 2 (a 1/2 exponent means square root)
\n" ); document.write( "You can raise 4 to the -1 power to get 1/4
\n" ); document.write( "etc.
\n" ); document.write( "Many of these \"right powers\" are difficult to know. For example what power of 4 results in a 5? What power of 13 results in 1/3? etc.

\n" ); document.write( "Logarithms are used to express these exponents, both the well-known and the unknown. In general $\"log%28a%2C+%28p%29%29\"$ represents the exponent you use on \"a\" to get \"p\". Some specific examples:
\n" ); document.write( " $\"log%284%2C+%2816%29%29\"$ is the exponent for 4 that results in 16. We know this one. It is 2. So $\"log%284%2C+%2816%29%29+=+2\"$ .
\n" ); document.write( " $\"log%28100%2C+%2810%29%29\"$ represents the exponent for 100 that results in 10. We know this one, too. It is 1/2. So $\"log%28100%2C+%2810%29%29+=+1%2F2\"$
\n" ); document.write( " $\"log%2841%2C+%2864%29%29\"$ represents the exponent for 41 that results in 64. This one we don't know.

\n" ); document.write( "So the logarithm in your expression, $\"log%283%2C+%288%29%29\"$ represents the exponent for 3 that results in 8. This is another one we don't know. But look at where it is! It iis the exponent on a 3! So your expression is raising 3 to the power which, when it is the exponent on a 3, which results in 8. IOW, your expression is 3 to the power for 3 that results in 8. So by the very definition of what logarithms mean:
\n" ); document.write( " $\"3%5Elog%283%2C+%288%29%29+=+8\"$ !!
\n" ); document.write( "In general
\n" ); document.write( " $\"a%5Elog%28a%2C+%28p%29%29+=+p\"$ for all valid a's and p's. \n" ); document.write( "

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