SOLUTION: use the properties of logarithms to elimate products, quotients and exponents from the following expression: log3[9 sqrt(x-5)/(x-1)^(5)] Thank you

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: use the properties of logarithms to elimate products, quotients and exponents from the following expression: log3[9 sqrt(x-5)/(x-1)^(5)] Thank you      Log On


   

Question 387318: use the properties of logarithms to elimate products, quotients and exponents from the following expression: log3[9 sqrt(x-5)/(x-1)^(5)]
Thank you
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%283%2C+%28%289%2Asqrt%28x-5%29%29%2F%28x-1%29%5E5%29%29
First we can use the property for quotients, log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29, to split the single logarithm into the difference of the logarithms of the numerator and denominator:
log%283%2C+%289%2Asqrt%28x-5%29%29%29+-+log%283%2C+%28%28x-1%29%5E5%29%29
On the first logarithm we can use the property for products to split the logarithm of the product into the sum of the logarithms of its factors:

Since square roots are the same as an exponent of 1/2 we can rewrite the second logarithm as a power of x-5:

Now we can use the property for exponents to rewrite the last two logarithms:
log%283%2C+%289%29%29+%2B+%281%2F2%29log%283%2C+%28x-5%29%29+-+5%2Alog%283%2C+%28x-1%29%29
Last of all the first logarithm is one we can find (without a calculator). Since 9+=+3%5E2 then the first logarithm must be 2:
2+%2B+%281%2F2%29log%283%2C+%28x-5%29%29+-+5%2Alog%283%2C+%28x-1%29%29
This is as far as we can go.