Expert: Ahmed Salami - 11/14/2008

Question
QUESTION: How do I simplify:
7log base7^4 - ^2log base7^2 + 3log base3^5 + log base3^2 - 2 ln e^3?

I believe the second part simplifies to 10, but I'm confused about the first and last parts.
Thanks for your help!

ANSWER: Hi,
I'm sorry Keli but the question seems unclear to me. Could you rewrite and send again.

Regards.

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

QUESTION: Please check this link to see a picture of the problem:

http://img.photobucket.com/albums/v199/emperorjordan/shane.jpg

Thank you!

Answer
Hi Keli,
You're right about the second part being 10 because
log(3) 5 + log(3) 2 = log(3) 5x2
         = log(3) 10
And,
3^[log(3) 5 + log(3) 2] = 3^log(3) 10
         = 10
In the same way,
log(7) 4 - 2log(7) 2 = log(7) 4 - log(7) 2^2
         = log(7) 4 - log(7) 4 = 0
(because a.log x = log (x^a) whatever the base is, so base doesnt appear in the relation)
7^[log(7) 4 - 2log(7) 2] = 7^0 = 1

2ln(e^3) = ln(e^3)^2
        = ln(e^6)
        = 6    (because ln x = log(e) x)
Therefore,
7^[log(7) 4 - 2log(7) 2]  +  3^[log(3) 5 + log(3) 2]  -  2ln(e^3)
= 1 + 10 - 6
= 5

Regards