document.write( "Question 718223: find the value of log125^25 \n" ); document.write( "
Algebra.Com's Answer #440820 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
  • The expression,as you posted it, is \"log%28%28125%5E5%29%29\". If this is correct, then just type it into your calculator.
  • If the expression is: \"log%28125%2C+%285%29%29\" then...
    • There is no exponent in this expression, even though it looks there is. The second 5 is not an exponent of the 125!! The 125 is the base of the logarithm and the 5 is the argument of the logarithm. Posting logarithms are difficult. To post a log like this, either
      • Use some English to help describe the log. For example, this one could be described as: \"the base 125 log of 5\"; or...
      • Teach yourself the syntax we use on algebra.com to get nice-looking logs like \"log%282%2C+%2815%29%29\". Click on the \"Show source\" link above to see what I've typed to get these logs to look good.
    • The expression \"log%28125%2C+%285%29%29\" represents the exponent one would put on 125 to get a result of 5. So this problem is asking us to figure out: What power of 125 is 5? Perhaps you already know the answer. Perhaps knowing that \"5%5E3+=+125\" would help you figure out the answer. (It's not 3. 3 is the exponent we put on 5 to get 125. We're looking for the exponent to put on 125 to get 5.) If you still don't know the answer, then we can use \"5%5E3+=+125\" to figure it out. We will use the change of base formula, \"log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29\" to convert the base 125 log to an expression of base 5 logs:
      \n" ); document.write( "\"log%28125%2C+%285%29%29+=+log%285%2C+%285%29%29%2Flog%285%2C+%28125%29%29\"
      \n" ); document.write( "The numerator of this fraction is the exponent you put on a 5 to get 5. This should be 1, obviously. The denominator is the exponent you put on a 5 to get 125. From \"5%5E3+=+125\" we know that this exponent is 3. So now we have:
      \n" ); document.write( "\"log%28125%2C+%285%29%29+=+log%285%2C+%285%29%29%2Flog%285%2C+%28125%29%29+=+1%2F3\"
      \n" ); document.write( "I hope this answer makes sense now. An exponent of 1/3, remember, means cube root.
    \n" ); document.write( "
\n" );