SOLUTION: If log(3,x) = 8, evaluate the following. log(3,9x) I was given the answer 10 but I don't know the reasoning.

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Question 1204575: If log(3,x) = 8, evaluate the following.
log(3,9x)

I was given the answer 10 but I don't know the reasoning.
Found 3 solutions by mananth, greenestamps, ikleyn:
Answer by mananth(16949) About Me  (Show Source):
You can put this solution on YOUR website!


log(3,x) = 8
3^8=x=6561
log(3,9x)
9x= 9*6561=59049=3^10
9x=3^10
log(3,59049) = log(3,(9x))=10



Answer by greenestamps(13292) About Me  (Show Source):
You can put this solution on YOUR website!


There is no need to work with the actual powers of 3, as the other tutor did, to work this problem....

The solution is in a single line, using basic log rules (and no big numbers!):

Answer by ikleyn(53560) About Me  (Show Source):
You can put this solution on YOUR website!
.
If log(3,x) = 8, evaluate the following.
log(3,9x)
I was given the answer 10 but I don't know the reasoning.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~


            The most simple reasoning is  THIS : 


    log%283%2C%289x%29%29 = log%283%2C%289%29%29 + log%283%2C%28x%29%29 = 2 + 8 = 10.    ANSWER


On the way, I used two facts:  

        log%283%2C%289%29%29 = log%283%2C%283%5E2%29%29 = 2%2Alog%283%2C%283%29%29 = 2*1 = 2,

    and

        log%283%2C%28x%29%29 = 8  (given).

Solved.

This solution shown in my post is what the problem and the teacher do expect from you.

Comparing with the solution by @mananth, the advantage of this solution is
that I do not make no one excessive calculation and use the most straightforward economic way.

In whole, this my composition is "the most elegant and the most expected/desired way to solve".