Expert: Frederick Koh - 12/12/2011

Question
QUESTION: Find the derivative of ln[ln(x^2)^5].

Please show steps. Thanks!

ANSWER: Firstly, we will try to transform the logarithm for simpler outlook:
ln[ln(x^2)^5]= ln [ ln(x^10)] =ln ( 10 ln x )

Hence,

d/dx[ ln ( 10 ln x )] = (10/x)/ (10 ln x) = (1/x)/(ln x) = 1/ (x ln x)      (shown)

Hope this helps. Peace.

---------- FOLLOW-UP ----------

QUESTION: My solution manual said the answer was 5/xlnx.  Do you think I typed the problem down wrong or is it supposed to be ln[5ln(x^2)]to get the answer of 5/xlnx.  Sorry for the confusion.

ANSWER: If your solution manual states that 5/(xlnx) is the answer, the only way you are going to get that is if the question asks you to find the derivative of ln{[ln(x^2)]^5}  . Note that this is totally different from ln[ln(x^2)^5] or  ln[5ln(x^2)] which you suggested in this follow-up.

The placement of brackets to a certain extent may( or may not) influence the actual interpretation of the logarithm in question, hence you have to be exceptionally mindful of that.

Hope this helps. Peace.

---------- FOLLOW-UP ----------

QUESTION: Thank you! Could you please show me the steps to solve ln{[ln(x^2)]^5}.  I keep getting 10/xln(x^2).

Answer
Sure. Here goes:

ln{[ln(x^2)]^5} = 5 ln [ln(x^2)]   (Note that I pushed the 5 all the way to the front)
         = 5 ln [ln(x)^2)]  
         = 5 ln [ 2ln(x) ]    [I do something similar for the 2 on the inside as well)

Differentiating  5 ln [ 2ln(x) ] with respect to x gives  
      
     5 [ (2/x) / (2lnx) ] = 5 [ (1/x) / (lnx) ] = 5[ 1/(xlnx)] =5/(xlnx)

Hope this helps. Peace.