Question 354508
{{{log(a, (2)) = p}}}
{{{log(a, (3)) = q}}}
{{{log(a, (sqrt(6)))}}}
The first key to this problem is to know that square roots can be expressed using fractional exponents: {{{sqrt(x) = x^(1/2)}}}. So we can rewrite your expression using an exponent of 1/2:
{{{log(a, (6^(1/2))))}}}
Now we can start using the properties of logarithms:<ul><li>{{{log(n, (x*y)) = log(n, (x)) + log(n, (y))}}}</li><li>{{{log(n, (x/y)) = log(n, (x)) - log(n, (y))}}}</li><li>{{{log(n, (x^y)) = y*log(n, (x))}}}</li></ul>
We can use the third property to remove the exponent:
{{{(1/2)*log(a, (6))}}}
Now we need to change the argument from a six to something in terms of 2's and/or 3's. It should be obvious that 6 = 2*3 so now we have:
{{{(1/2)*log(a, (2*3))}}}
In this form we can use the first property to separate the log into two logs:
{{{(1/2)*(log(a, (2)) + log(a, (3)))}}}
Next we can subsitute in for the two logs:
{{{(1/2)*(p + q)}}}
And finally we can use the Distributive Property:
{{{(1/2)*p + (1/2)*q}}}