Question 387318
{{{log(3, ((9*sqrt(x-5))/(x-1)^5))}}}
First we can use the property for quotients, {{{log(a, (p/q)) = log(a, (p)) - log(a, (q))}}}, to split the single logarithm into the difference of the logarithms of the numerator and denominator:
{{{log(3, (9*sqrt(x-5))) - log(3, ((x-1)^5))}}}
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:
{{{log(3, (9)) + log(3, (sqrt(x-5))) - log(3, ((x-1)^5))}}}
Since square roots are the same as an exponent of 1/2 we can rewrite the second logarithm as a power of x-5:
{{{log(3, (9)) + log(3, ((x-5)^(1/2))) - log(3, ((x-1)^5))}}}
Now we can use the property for exponents to rewrite the last two logarithms:
{{{log(3, (9)) + (1/2)log(3, (x-5)) - 5*log(3, (x-1))}}}
Last of all the first logarithm is one we can find (without a calculator). Since {{{9 = 3^2}}} then the first logarithm must be 2:
{{{2 + (1/2)log(3, (x-5)) - 5*log(3, (x-1))}}}
This is as far as we can go.