SOLUTION: If a=log(base7)(11-6√2) and b=log(base7)(45+29√2), find 3a+2b in the simplest form.

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: If a=log(base7)(11-6√2) and b=log(base7)(45+29√2), find 3a+2b in the simplest form.      Log On


   

Question 253720: If a=log(base7)(11-6√2) and b=log(base7)(45+29√2), find 3a+2b in the simplest form.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
If a=log%287%2C%2811-6%2Asqrt%282%29%29%29 and b=log%287%2C%2845%2B29%2Asqrt%282%29%29%29, find 3a+2b in the simplest form.

a+=+log%287%2C%2811-6%2Asqrt%282%29%29%29 if and only if 7%5Ea+=+11-6%2Asqrt%282%29

b+=+log%287%2C%2845%2B29%2Asqrt%282%29%29%29 if and only if 7%5Eb+=+45%2B29%2Asqrt%282%29

take log of both sides of first equation to get:

7%5Ea+=+11-6%2Asqrt%282%29 becomes:

log%28%287%5Ea%29%29+=+log%28%2811-6%2Asqrt%282%29%29%29

this becomes:

a%2Alog%28%287%29%29+=+log%28%2811-6%2Asqrt%282%29%29%29

divide both sides by log((7)) to get:

a+=+log%28%2811-6%2Asqrt%282%29%29%29%2Flog%28%287%29%29 *************

take log of both sides of second equation to get:

7%5Eb+=+45%2B29%2Asqrt%282%29 becomes:

log%28%287%5Eb%29%29+=+log%28%2845+%2B+29%2Asqrt%282%29%29%29

this becomes:

b%2Alog%28%287%29%29+=+log%28%2845+%2B+29%2Asqrt%282%29%29%29

divide both sides by log((7)) to get:

b+=+log%28%2845%2B29%2Asqrt%282%29%29%29%2Flog%28%287%29%29



since the denominator is the same, we get:



I don't think you can simplify it any further, but you can solve it using the LOG function of your calculator.

equation becomes:

3a+%2B+2b+=+%283%2A.400489398+%2B+2%2A1.934560022%29%2F.84509804

this becomes:

3a+%2B+2b+=+%281.201468195+%2B+3.869120045%29%2F.84509804+=+6

I confirmed the answer is correct by using the logarithm base conversion formula to convert base 7 logarithm to a base 10 logarithm.

that conversion formula is:

log%287%2C%28x%29%29+=+log%2810%2C%28x%29%29%2Flog%2810%2C%287%29%29

you could also have solved this problem directly by using the base conversion formula.