SOLUTION: Express as a single logarithm, and, if possible, simplify?? 4 loga y + 1/7 loga z Thank you for your help!

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Question 250423: Express as a single logarithm, and, if possible, simplify??
4 loga y + 1/7 loga z
Thank you for your help!
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the rules of logarithms state:

log%28a%2Ab%29+=+log%28a%29+%2B+log%28b%29 (rule 1)

log%28a%2Fb%29+=+log%28a%29+-+log%28b%29 (rule 2)

log%28a%5Eb%29+=+b+%2A+log%28a%29 (rule 3)

your expression is:

4+%2A+log%28a%2Cy%29+%2B+%281%2F7%29+%2A+log%28a%2Cz%29

4+%2A+log%28a%2Cy%29+=+log%28a%2Cy%5E4%29 by rule 3.

%281%2F7%29+%2A+log%28a%2Cz%29+=+log%28a%2Cz%5E%281%2F7%29%29 by rule 3.

your expression becomes:

log%28a%2Cy%5E4%29+%2B+log%28a%2Cz%5E%281%2F7%29%29

this becomes log%28a%2C%28y%5E4%2Az%5E%281%2F7%29%29%29 by rule 1.

since z%5E%281%2F7%29 can also be written as root%287%2Cz%29, this can also be written as:

log%28a%2C%28y%5E4%2Aroot%287%2Cz%29%29%29

you can prove that your original expression of 4+%2A+log%28a%2Cy%29+%2B+%281%2F7%29+%2A+log%28a%2Cz%29 is the same as your final expression of log%28a%2C%28y%5E4%2Aroot%287%2Cz%29%29%29 by substituting for the values of a, y, z.

I used a = 10 because this base is solvable by using the LOG function of the calculator.

I let y = 7
I let z = 15

I got the same answer for the original expression and the final expression which says that I made the translation correctly.